LIE GROUPS: HISTORY, FRONTIERS
AND APPLICATIONS
VOLUME XIII
Geometry of Riemannian
Spaces
BY ELIE CARTAN
TRANSLATED BY JAMES GLAZEBROOK
NOTES AND APPENDICES
BY R. HERMANN
ΙϊΙΔΰΗ ЪС1 ?ЧкЪЪ

LIE GROUPS: HISTORY, FRONTIERS
AND APPLICATIONS
VOLUME XIII
GEOMETRY OF RIEMANHIAN SPACES
BY ELI Ε CARTAN

LIE GROUPS: HISTORY FRONTIERS AND APPLICATIONS
Note: This series has outgrown its original plan, hence it will now
bifurcate. SERIES A will continue the series of translations of the classics.
SERIES A
1. Sophus Lie's 1880 Transformation Group Paper. Translation by M.
Ackerman, Comments by R. Hermann
2. Ricci and Levi Civita's Tensor Analysis Paper, Translation and
Comments by R. Hermann
3. Sophus Lie's 1884 Differential Invariants Paper, Translation by M.
Ackerman, Comments by R. Hermann
4. Smooth Compactification of Locally Symmetric Varieties, by A. Ash,
D. Mumford. M. Rapoport and Y. Tai
5. Symplectic Geometry and Fourier Analysis, by N. Wallach
6. The 1976 Ames Research Center (NASA) Conference on The
Geometric Theory of Non-Linear Waves
7. The 1976 Ames Research Center (NASA) Conference on Geometric
Control Theory
8. Hubert's Invariant Theory Papers. Translation by M. Ackerman,
Comments by R. Hermann
9. Development of Mathematics in the 19th Century, by Felix Klein,
Translated by M. Ackerman, Appendix "Kleinian Mathematics from
an Advanced Standpoint," by R. Hermann
10. Quantum Statistical Mechanics and Lie Group Harmonic Analysis,
Part A, by N. Hurt and R. Hermann
11. First Workshop on Grand Unification by P. Frampton, S. Glashow
and A. Yildiz
12. Inverse Scattering Papers: 1955-1963 by I. Kay and Η. Ε. Moses
13. Geometry of Riemannian Spaces, by Elie Cartan. Translated by J.
Glazebrook, Commentary by R. Hermann
SERIES B: SYSTEMS INFORMATION AND CONTROL
1. Geometry and Identification, P. E. Caines and R. Hermann, Eds.

LIE GROUPS: HISTORY, FRONTIERS
AND APPLICATIONS
VOLUME XIII
GEOMETRY OF RIEMANNIAN SPACES
BY ELIΕ CARTAN
TRANSLATED BY JAMES GLAZEВROOK
NOTES AND APPENDICES BY R. HERMANN
MMHSCI PRESS
53 JORDAN ROAD
BROOKLINE, MASSACHUSETTS 02146

Translation of begone aur la Geometric
dee Sepaoea de Riemotm
Second Edition, 1951
Permission to translate given by Gauthler-Vl1lars
Publisher of French Edition
Copyright О 1983 by Robert Hermann
All rights reserved
Llbriry of Congress CiUleglng in Publicities DiU
Cartaa, Ше, 1869-1951.
Oeoeetiy of Riemannlan spaces.
(Lie groups ; т. 13)
Bibliography: p.
1. Oeoeetiy, Rienannian. I. Title. II. Series.
QA6U9-C313 1983 5l6.3'T3 83-11982
ISBN 0-915692-3U-1
ттн sci press
53 JORDAN ROAD
BROOKLINE, MASSACHUSETTS 02Й6
Printed In the United Stetes of America

PREFACE TO THE FIRST EDITION
This text follows a course given 1n the first semester 1925-1926 1n the
Faculty of Sciences of the University of Paris. I have adopted, in principle,
the same point of view as 1n the text on the same subject that 1s Included 1n
the series Memorial dee Sciences Mathemtztiquea (No. IX). I have almost always
assumed given a system of local coordinates and a metric tensor for the space
1n question: This has required Ideas from tensor analysis. I have tried to
do this while emphasizing the essential qeonetrlc features and keeping 1n
close contact with Euclidean oeometry. The utility of the absolute differential
calculus of R1cc1 and Lev1-C1v1ta must be tempered by an avoidance of
exclusively formal calculations, where the debauch of Indices disguises an
often very simple geometric reality. It 1s this reality that I have sought
to reveal.
I present an account of the Interesting problems of those spaces, which
while locally Euclidean, are from the point of view of "analyaie еъЬш'\
different from our ordinary space. This involves the "CHfford-Kle1n space
forms'* of the German school of geometry. The perspectives that one obtains on
the foundations of elementary geometry and on certain theorems 1n analysis, by
the solution of this problem, seem to me to justify their status here. For
similar reasons, I have looked closely at the Important role played 1n geometry
by the axiom of the plane and the axiom of free nobility, each Intimately
linked to the other. This leads me quite naturally to review the non-Euclidean
geometries, particularly 1n two dimensions.
The first two Notes that appear at the end of the text foil ом up certain
Ideas pursued 1n the course of the book, via the Introduction of hypotheses
far less restrictive on the analytical nature of the coefficients of the
fundamental form. In this respect I believe that the concept of linear (as
opposed to euperfiaial) Rlemannian curvature had not been described; It would
undoubtedly have applications 1n relativity theory. The third Note, devoted
to spaces with variable negative curvature, relates to the now classical
paper by N. Hadamard, on the geodesies of surfaces of negative curvature.
The bibliography at the end of the book serves to Indicate the original
texts and papers relating to the topics discussed.
In this book I have put to the side many Important problems. They will
perhaps be the aim of a further work, where the rethod of the moving
rectangular frame would be dealt with 1n greater depth, together with Its
numerous applications.

In concluding, I would like to express my thanks to Mai son Gauthier-Viliars
(ho published this work with their customary care and attention.
E. Carta η

PREFACE TO THE SECOND EDITION (1951)
This new edition differs from the first by sane amendments and additions,
the most Important of which I shall mention.
I have adopted the now classical convention due to Einstein, which entails
dispensing with the sign of summation wherever there 1s no possible ambiguity.
The most Important modification 1s the replacement of ω' to denote what I
would have called the "exterior derivative" of the differential form ω, by a
more satisfactory notation ώα; this 1s due to E. KShler and denotes what I
now cell the "exterior differential11 of ω. The symbol D now serves to
denote absolute differentiation, for which the operand 1n question 1s an
ordinary tensor or a tensor differential form, hence a unifying notation which
would be more appropriate.
The most Important additions appearing 1n this second edition fall Into
three sections. Firstly, I have Introduced a new chapter on the nethod of the
moving frame, with applications to the properties of manifolds embedded 1n a
R1emann1an space. Secondly, the chapter on normal R1emann1an co-ordinates
(the original Chapter IX) has been almost halved by the Inclusion of a new
chapter (XI) on symmetries, parallel transport and symmetric spaces. Finally,
two new chapters (XII and XIII) of which the first 1s quite extensive, are
devoted to displacement groups in Rienwnnion space and the mapping conditions
for two R1emann1an spaces, respectively. Both these new chapters Involved
using the method of the moving frame almost exclusively.
At the end, two new Notes have been added to three Notes of the first
edition. One (Note IV) 1s devoted to the properties of geodesies 1n a normal
R1emann1an space, the other (Note V) to completely Integrable Pfafflan systems.
Despite the present difficulties, Halson 6auth1er-V1liars have given all
of their customary care to the publishing of this new edition. I am most
grateful to them.
E. Cart an

PREFACE BY ROBERT HERMANN
As I have worked my way Into 20th century geometry with my comments 1n
this series of translations, I have wanted to do Cartan. He 1s the key figure
Unking the world of 19th century and post 1950 differential geometry. Indeed,
much of our collective effort since 1950 has gone Into working out Ideas
originated by Cartan or develooed by him on the foundation of the work by his
19th century predecessors.
Although 1t 1s not the book that 1s most central to his own work (I would
nominate Loo Syatwneo Difforantioltaa Eartai^iaiure at taurv Applioatione
Geometriqu&B for that honor), I believe that Geonetrie dee Еврасея de Riemcmn
1s the best place to begin, both 1n terms of the historical structure and the
needs of a student today who wants an Introduction to Cartan's methods. The
First Edition was written at the peak of popularity of tensor analysis, and
Cartan made efforts to show how his methods compare with those of his
contemporaries. It 1s also the most carefully written of his books from the
pedagogical viewpoint. I read 1t and found 1t most valuable when I began my
study of differential geometry. Finally, as another sign of Cartan's uncanny
foresight, this 1s really the first place where the global and topologioal
problems of R1emann1an geometry were treated. (In this period, Cartan was
beginning, following partially th· lead of Hermann Ueyl, to work on global
and topological problems of Lie grouo and symmetric space theory.)
Thus, I hope that James Glnzebrook's efforts 1n preparing this translation
will make some of Cartan's magnificent geometric methodology more accessible
to a new generation of "pure" and "apolled" mathematicians, physicists, and
engineers who are just now developing many of the Ideas In Cartan's work.
I would like to thank Joyce Martin and Karl η Young for typing, and
Greg Ammar for help with proofreading.

TRANSLATOR'S ACKNOWLEDGEMENTS
This translation of EHe Cartan's Leqons but la GSomStrie dee Eepaoee de
Riemcmn Mas completed 1n the academic year 1978-79.
I am deeply grateful to Dr. P. DoIan of the Mathematics Department of
The Inperlal College of Science and Technology, London, for hevlng the Insight
to suggest that the re-1ntroduct1on of Cartan's text, would be a worthwhile
project. His conwents, criticisms and hearty encouragement 1n the early stages
of the translation were of great value to me, as ware the elements of his
lecture course on the theory of Lie Groups that I pursued the year before.
My thanks extend to Kathleen Akerman and Hary Williams who both helped to
produce a splendid typescript and also to dear friends Bryson and Margaret who
regularly offered their hospitality so thet I might peacefully proceed with
the work 1n question.
James Glazebrook
Barnes, London
August 1979

TABLE OF CONTENTS
PREFACE TO THE FIRST EDITION
PREFACE TO THE SECOND EDITION
PREFACE BY ROBERT HERMANN
TRANSLATOR'S ACKNOWLEDGEMENTS
Page
111
ν
v11
1x
Chapter 1: CARTESIAN CO-ORDINATES: VECTORS, HULTIVECTORS, TENSORS
I. Vectors, Cartesian co-ordinates
II. B1vectors, systems of Μ vectors
III. Tr1vectors
IV. Hultl vectors
V. Supplementary multlvectors
VI. Sliding or bound multl vectors
VII. Application to the motion of a rigid botfy having
a fixed point
VIII. Tensors, tensor algebra
1
1
4
11
14
15
18
19
20
Chapter 2: CURVILINEAR CO-ORDINATES IN EUCLIDEAN GEOMETRY 29
I. The spatial line element 1n Cartesian co-ordinates 29
II. The fundamental theorem of Metric geonetry 31
III. The local reconstruction of a space 1n terms of Its
line element 34
IV. Absolute differentiation. K1nemat1ca1 applications.
Lagrange's equations 37
V. Tensor analysis 43
VI. The necessary condition to be satisfied by the Euclidean
line element 48
VII. Euclidean line elements 52
Chapter 3: LOCALLY EUCLIDEAN RIEKANNIAN SPACE
I. The concept of a manifold
II. Locally Euclidean R1emann1an space
III. Normal locally Euclidean R1enann1an space
IV. The holonomy group of a normal locally Euclidean
R1emann1an space
V. The fundamental polyhedron
VI. Determination of the normal locally Euclidean
R1eraann1an soaces
VII. Two dimensional normal Euclidean spaces
VIII. Normal locally Euclidean R1emann1an space and
elementary geometry
57
57
59
63
68
70
73
74
82
Chaoter 4:
I.
II.
III.
IV.
RIEHANNIAN SPACE AND EUCLIDEAN TANGENT AND OSCULATING
SPACE 85
The Euclidean tangent space at a point 85
The Euclidean osculating space 89
The Euclidean space along a line 99
Application to the theory of surfaces in ordinary space 105
x1

x11
Chapter 5:
I.
II.
III.
GEODESIC SURFACES. THE AXIOHS OF THE PLANE AND THE
AXIOM DF FREE NOBILITY
Geodesic surfaces at a point. Severl's Theorem
Totally geodesic surfaces. Planes
The axiom of the Diane and the axiom of free mobility
in space
CONTENTS
Page
HI
HI
112
116
Chanter 6: NON-EUCLIDEAN GEOMETRIES. SPHERICAL SPACE. ELLIPTIC
SPACE. HYPERBOLIC SPACE 125
I. Two dlnenslonal spherical geowetry 125
II. Two dlnenslonal elliptic geoeetry 126
III. Two dimensional hyperbolic geooetry 133
IV. Conformal representation of spherical and hyperbolic
geometries 137
V. The displacement group of the non-Euclidean geometries 146
VI. Three dimensional non-Euclidean space: projective
representation 149
VII. Three dlnenslonal non-Euclidean space: conformal
representation 157
VIII. Normal R1enann1an space, the locally spherical and
hyperbolic cases 162
IX. Three dlnenslonal R1emann1an space satisfying the
axiom of the plane 168
Chapter 7: RIEMANNIAN CURVATURE
I. The dlsplacenent associated with a cycle
II. The R1emann-Chr1stoffel tensor
III. R1emann1an curvature 1n two dlnenslonal space
IV. R1emann1an curvature 1n three dlnenslonal space
V. R1emann1an curvature 1n spaces of more than three
dlnenslons. Spaces with constant R1emann1an curvature
VI. The contracted curvature tensor. Principal directions
173
173
178
180
185
191
196
Chapter B: THE ВIAHCHI IDENTITIES
I. Exterior differential forms
II. Differential tensor forms
III. The Bianchl Identities
IV. Polncare's theorem 1n R1enann1an space
V. Curvature vectors and their first representation
VI. Curvature vectors and their second representation
VII. Schur's theorem
197
197
204
206
207
209
212
214
Chapter 9:
I.
II.
III.
IV.
THE METHOD OF THE MOVING FRAME. MANIFOLDS EMBEDDED IN
THREE DIMENSIONAL RIEMANNIAN SPACE 217
Generalities 217
Completions to the theory of surfaces embedded 1n three
dimensional R1emann1an space 219
Curvature lines and asymptotic lines of a manifold
embedded 1n a R1emann1an space 225
Riemannian spaces that satisfy the axiom of the plane 228

CONTENTS
x111
Chapter 10: NORMAL RIEMANNIAN CO-ORDINATES
I. Normal co-ordinates
II. The fundamental differential eouatlons
III. The ds2 of spaces with constant curvature expressed
1n nomal co-ordinates
IV. Properties of the fundamental form 1n normal co-ordinates
V. Comparison of distances 1n R1enann1an space and 1n normal
osculating Euclidean space
VI. The parallelogramold of Lev1-C1v1ta
VII. Geodesic triangles
VIII. Circles, spheres and hyperspheres
Page
231
231
232
236
238
241
244
245
249
Chapter 11: SYMMETRIES AND PARALLEL TRANSPORT. SYMMETRIC SPACES 255
I. Symmetry and parallel transport 255
II. Symmetric R1eroann1an spaces 260
III. Rigid displacements of a symmetric space 263
IV. Irreducible symmetric spaces 264
Chapter 12: RIGID DISPLACEMENT GROUPS IN RIEMANNIAN SPACE 269
I. Generalities 269
II. Transitive and Intransitive groups. Trajectories 270
III. Frames attached to a displacement group 271
IV. R1eroann1an spaces admitting a simply transitive
displacement group 273
V. Canonical co-ordinates 1n a space admitting a simply
transitive displacement group 277
VI. Canonical co-ordinates and normal co-ordinates 280
VII. Isogonal parallelism attached to a simply transitive
displacement group 234
VIII. R1emann1an spaces admitting a multiply transitive
displacement group 290
IX. Three dimensional space admitting a multiply transitive
group 296
X. General Intransitive displacement groups 303
XI. Displacement groups whose trajectories are lines or
surfaces 306
Chapter 13: ISOMETRIC RIEMANNIAN SPACES. RIGID DISPLACEMENTS IN A
GIVEN SPACE 309
I. Isometric R1emann1an spaces 309
II. An analytical problem 312
III. The general problem concerning the mapping of
Riemannlan spaces 315
IV. The maximum displacement πroup of a given Riemannlan
space 322
V. Killing's equations 324
NOTE I: On the Axiom of the Plane and Cayleylan geometry
NOTE II: On Linear Riemannlan Curvature
327
335
NOTE III:
On Normal Spaces with Zero or Negative Riemannlan
Curvature
339

CONTENTS xiv
Page
NOTE IV: Geodesies of Normal R1enann1an Spaces 355
NOTE V: Completely Integrable Pfafflan Systems 365
BIBLIOGRAPHY 371
INTRODUCTION BY ROBERT HERMANN 373
NOTES ON "GEOMETRIE DES ESPACES DE RIEKANN" 375
APPENDIX 1: THE FORMALISM OF CONNECTION THEORY 377
APPENDIX 2: CARTAN'S METHOD OF THE MOVING FRAME AS A GENERALIZATION
OF KLEIN'S "ERLANGER PROGRAM" 457
APPENDIX 3: EXCURSIONS INTO THE THEORY OF CARTAN CONNECTIONS 489

Chapter I
CARTESIAN CO-ORDINATES: VECTORS, MULTIVECTORS, TENSORS
I. VECTORS, CARTESIAN CO-ORDINATES
1. We assume that the classical theorems on vector addition are known.
We recall that if a vector 1s Indicated as £, the symbol mx_, where m 1s
a numerical coefficient, denotes a vector parallel to x^ whose length 1s m
times that of x^ and whose sense of direction 1s the same as that of £ or
the opposite, depending on whether m 1s positive or negative, respectively.
Let us also recall that if rectangular axes are taken 1n Euclidean space
of n-d1mens1ons, the square of the length of a vector x^ whose components are
1' 2 · ■ ■ ■ · *и» ' ^
X? + X? + ... + X2
1 2 η
The scalar product of two vectors x_ and / with respective components X.
and Y, 1s similarly
X·
i'i-VnV!*- +ΧΛ ·
We note that this scalar product could be obtained as the coefficient of 2λ
1n the expressions giving the square of the length of the vector χ. + λ^
(χ + λν)2 =- χ2 + 2λ χ · у + λ2 = Σ Χ? + 2λ Σ ΧνΥν + λ2 Vy?
■ι1 ΐ11 ΐι
The scalar multiplication of two vectors is cormutative and distributive, and
1s expressed as
*.' X. " Z'*. (*_+£) · L " *.*! + /" A respectively.
The cosine of the angle θ between vectors x^ and / 1s then given to be
^гЧЧ
(1)
ъ τ»
2. In ordinary space we obtain the most general system of Cartesian
co-ordinates by taking three co-ordinate axes Oar., Or.» 0*3 and defining on
each of them a unit of length. The co-ordinates of a point Μ are then taken
to be the algebraic magnitudes of the projections of the vector OM on these
axes, each vector being measured 1n terms of the unit of length defined on the
1

2
Cartesian Co-ordinates
corresponding axis, and the projection being made on each axis parallel to the
plane determined by the other two axes.
These systems of co-ordinates, more general than those usually considered
1n analytic geometry, evidently possess the property that a change 1s made from
one of these systems to another by means of a linear transformation with
constant coefficients. Effectively, by letting S,, SL, 2, be the co-ordinates
of the origin of the new system with respect to the original, let us denote by
(o'xp
(o'x2')
(o'Y*
all
a12
a13
a21
a22
a23
a31
a32
a33
the respective projections on the original axes of the unit vectors given on
the new axes.
The passage of the co-ordinates (r'.r'.r') of a point with respect to
the new system to the co-ordinates (г,,χ-,λ,) of a point with respect to the
original, 1s given by the equations
rl "Sl *β11*ί +α1Ζ*2 + α13*3
x2 * S2 + a21ri + altl + a23r3
r3 " S3 + a31ri + a32x2 + a33r3 *
Conversely, every group of relations of the above type defines a change 1n
Cartesian co-ordinates 1f the determinant of the (a.,) 1s always nonzero,
and 1t 1s possible to determine arbitrarily the original system of
coordinates.
In the case of a vector, the equations reduce to
Xl " α11Χί + αη4 +α13Χ3
X2 - a21Xf + a22X2 + а23Хз (det a^ r 0)
X3 - а^Ц + a32X2 + аггГг
where X,, X?, X, denote the projections of the vector on the original axes
and XC, XX· X3 their projections on the new axes.
These considerations may be generalised to n-d1mens1onal space without
any difficulty.
3. The components of a vector 1n any system of Cartesian co-ordinates
are derived from a homogeneous linear transformation between Its components 1n
a system of rectangular co-ordinates, where the square of the length of a

Cartesian Co-ordinatee
3
vector with components X,...X will be given by a quadratic form
■■J
Knowing the coefficients [g.,) of this form, allows us to form the
scalar product of two vectors x^ and £. Effectively, we have:
from which
t , J
(3)
Out of necessity, we assume that the Indices it jt take all possible values
Independently of each other on the right hand side. Hence the cosine of the
angle between two vectors 1s deduced to be
Σ,9·- x-Y-
Cos V - Ч г 3 (4)
The coefficients {д..) may 1n fact be geometrically Interpreted as the
palrwlse scalar products of the unit vectors taken on the axes. Effectively,
1f these vectors [baeie vectors) are denoted by β,»£2»-·-»β , where the
vector e. has all Its components zero save the 1-th which 1s equal to 1,
then we have on applying the relation 1n (3),
£*■%■»« ■ (5)
4. From now on we shall write the components of a vector with raised
Indices as
χ - ΧΊβΊ + X2e, + ... + Xne
— —1 ^2 -n
Following the style adopted by G. R1cc1 and L. Lev1-C1v1ta, we Introduce η
new quantities X. (not to be confused with those represented by this notation
1n the preceding sections). These X. will be by definition, the scalar pro-
*■
duct χ · e. of the given vector χ with the unit vector е.. On account of
(3), we have
X1 " <?iktk {i ' Ί'2 η)· (6)
By following the now classical convention, we shall dispense with the sign of
summation Σ on the right hand side. It 1s therefore understood that a sum

4
Cartesian Co-ordinates
1s to be made of all the terms obtained, by giving к the successive values
1,2,...,«. We will observe that the Index к occurs twice 1n the general
term on the right hand side; 1n one case as a lower Index and 1n the other
case» as an upper Index.
By the Introduction of these new quantities X. that are called the
1 i
oovariant components of the vector as opposed to the ordinary components X ,
known as the contravariant components, the expression for the scalar product
of two vectors x^ and / can be stated without distinction as one of the
two basic forms:
x-y = X,Yl = XlY.
and for the square of the length of a vector, we have
|x|2 * x.x^ .
(7)
(8)
The covarlant components of a vector are inversely related to their contravarl-
ant counterparts by the Identity
ХЪ IK u
= 3 h
(9)
where the gbJ is the minor relative to g.,, of the determinant
ЪГ
*11 ?12 "■ 3U
2*1 2nt '·· 3nn
this minor being divided by the value of the determinant Itself which we shall
denote by g.
A vector will be seen to have all Its contravarlant components zero save
the i-th when 1t is parallel to the г-th co-ordinate axis. On the other hand,
a vector having all Its covariant components zero save the г-th 1s
perpendicular to all the co-ordinate axes save the г-th, that is to say, perpendicular
to the {η-λ)-plane determined by these (n-1) axes.
We need not consider the case of rectangular co-ordinates where the
covarlant and contravarlant components are Identified with each other.
II. BIVECTORS, SYSTEMS OF BIVECTORS
5. А Ыvector 1s defined to be the configuration formed by two vectors
^ and /, arranged in a certain order. This definition only becomes
meaningful 1f the equality of two bivectors can be defined.
For this purpose, let us associate to а Ыvector, a parallelogram OACB
whose sides OA and 08 are equipollent to the vectors ^ and ^

Cartesian Co-ordinatee
5
respectively, with the perimeter of the parallelogram starting Its course at
OA. With this established, two Ыvectors are said to be equal 1f their two
associated parallelograms are 1n the same plane (or 1n parallel planes), have
the same area and sense of direction. We shall refer to the plane of the
b1vector as the plane of the associated parallelogram, or any parallel plane.
Let us thus define two b1 vectors, one by the two vectors x_, £, and
the other by the two vectors jj, _y_. The equations which give conditions for
a vector z. to be perpendicular to the plane of the first Μ vector are:
ΖΊΧΊ + Z9X2 + ... + Ζ Xя - 0
I С П
ζίυί + цч2 + ... + znY" - о
The components Ζ- ... Ζ of this vector are constrained by a unique relation
obtained by eliminating Ζ,, I.e.
z9(xV - xV) + z,(xV - xV) + ... + z (xV - χ"υί) - о
Let us put
pij B XV - x-V (10)
and similarly,
As the two equations Ζ.ΡΊι - 0 and Z.Ql1 - 0 are seen to be equivalent,
the quantities Ρ and Q l (г > Ί) are proportional to each other. In
p*J
practice, this Indicates that the ratio —ту does not change when one of the
Indices iJ is changed.
The first condition for equality of tuo biveatore ie thus represented by
a relationship of the form
Qij я xpij
It can also be seen that similar reasoning would lead to the relationship
Qij * μί4?
by putting
P.. - X.Y. - X.Y. and Q.. « U-V. - U.V. , (11)
and it 1s easily seen that λ ■ μ.

6
Cartesian Co-ordinatee
6. To express the second condition of equality* let us seek to evaluate
the area of the parallelogram OACB associated with the first given Μ vector.
The square of the area 1s
m2 > ОТ2 · UB2 S1n2 AOB - ОТ2 UB2 - (ОД · OB Cox AOB)2
.2
x2·*2 - (A"*.)2
On replacing the scalar products by their values, we have
X. X,
Y< V
xV - p.. xV
On noting that the sum P.. xV 1s equal to the sum P.. xV«-P.. J?V,
the required, definitive relationship 1s obtained:
m2 - *i P.. (xV - xV) ■ \ P.. P^ . (Ί2)
We shall note that for n*3 for example, the formula 1n (12) when expanded,
yields
m2 - P12P12 + P13P13+ P23p23
The second condition of equality of two blvectors, being a consequence of the
formula 1n (12), 1s that λ2 » 1, that 1s to say
Qij я ±p<j
The third condition (equality of direction) 1s realised by the ρresense of the
+ sign and not the - sign, and we arrive at th* following theoren:
For two biveotore to be equal, it ie neoeeeary and euffioient that the n*T^'
quantities Ρ J, defined by the relatione in (10) are the erne for the two
biveotore.
The quantities P1** will be known as the oo-ordinatee of the Ыvector.
The conditions of equality may also reduce to equality of the quantities P..,
i-i *'*
the oovariant component* of the Ыvector, the Ρ J being Its oontravariant
components.
Finally, we should bear 1n mind the relation in (12) which gives the
square of the magnitude of a b1vector.

Cartesian Co-ordinates
7
7. There 1s an alternative way of establishing the conditions of equality
of two Ыvectors: The condition expressing пом a vector ζ may be parallel
to the plane of the b1 vector defined by the vectors .x and y_, Js that ell the
determinants of three rows and col urns formed by the matrix
ΧΊΧ2
γ1γ2
should be zero.
obtained
... Z"
Xn
... Y"
In particular, a unique relation between Ζ
Ζ1 Ρ23 + Ζ2 Ρ31 + Ζ3 Ρ12 - 0
ι2, ζ2
1s
The result 1s that the equality of two Ыvectors Implies the relationship
.23
Ρ
Q3! Ql2
м
which simply means that the ratio ^W does not change when on* of the
Indices i,j 1s changed, whilst keeping the other fixed. All these ratios
are therefore equal between themselves.
Conversely, 1f we have
Q** - λΡ
a
the conditions stating how the vector г Is parallel to the plane of the
Μ vector are easily seen to be the same for the two bivectors.
With this established, we note that flji 1s simply the ratio of the
areas of the two parallelograms obtained by projecting onto the plane Ox,
0x2, the parallelograms associated with the two Ыvectors, the projections
naturally following the direction of the (n-2)-plane constituted by the axes
Ox.
Ox . The two Μ vectors will be equal 1f this ratio 1s equal to 1,
and vice-verse. The conditions of equality are thus stated as:
QiJ я Μ
(13)
8. The change from the contravarlant components P1** to Its covaHant
components P.. 1s not difficult to make:
We have
P..
13
X. X.
ι 3
h Ъ
gihX *J**
" *tA*p
hk

8
Car to вит Co-ordinat*e
It can be seen that the square of the magnitude of а Ыvector written as
* p% ■ «■ (»(Л, - щь) р*р* 04)
1s given by an expression quite like the one that gives the square of the
length of a vector (see relation (2)). The only difference 1s that here a
Ыvector has ui^li components Рг^ - -Ρ7* and that the coefficients of the
corresponding quadratic form are given 1n terms of the two composite iruHose
(£j) and (Wc), as
gHd)ihk) " Htfik ' 9itfjh
and furthermore, note that these are eyrmetrio:
9{ij){hk) -*<**>to"> ■
Moreover, the formation of the covariant component P. . 1n tenes of the con-
travarlant components P^ conies about in exactly the same way as for an
ordinary vector, by substituting the coefficients g,...,^. by the coefficients
Each bivector may therefore be represented by a vector in a Euclidean
space of η^~Ί' dimensions, the metric of which would then be defined by the
coefficients g{ij){hky
9. These results lead to the definition of the βσαίατ produot of two
Ыvectors in terms of any one of the two equivalent expressions:
4 t./fi * 4 PijQ.. · (15)
This scalar product evidently has a numerical value Independent of the choice
of co-ordinates. In order to interpret this geometrically, let us choose reo-
tangular axes, of which the first two are parallel to the plane of the first
given Μ vector. The components P,. - Ρ all being zero with the exception
of P,«, the scalar product will reduce to
p12 Q12 - (x,y2 - γ,ν (υ,ν2 - Vlu2) .
It ie evidently equal to the produot of the magnitude P,« of the firet biveo-
tor and the magnitude of the orthogonal projeotion of th* Beaond biveotor onto
the plane of the first. It 1s zero when there is a direction parallel to the
plane of one of the bivectors and perpendicular to the plane of the other.
It can be easily verified that, 1n ordinary 3-d1mens1onal space, the
scalar product of two Μ vectors is equal to the product of their magnitudes
and the cosine of the angle between their planes. This angle φ between the
planes of the two bi vectors a_ and b. may be defined by the formula

Cartesian Co-ordinates
9
Cos φ - ——— ;
the right hand side» as we can easily see. Is always less than or equal to 1
In absolute value.
10. The inner product of a blvector with components PtJ and a vector
with components Z1 Is the vector u^ with components
U* - Ρ*% . (16)
It Is Important to note that this vector has a significance Independent of the
choice of axes. Effectively, the scalar product of this vector u^ and an
arbitrary vector v^ Is
u4 - ^%h ' * ^ V* - ΖΛ> ·
It Is the scalar product of the given blvector and the blvector defined by z_
and v^ the scalar product Independent of the choice of axes. The product
u_· v^ Is Independent of the choice of axes, whatever the fixed vector v^ It
Is the same as the vector u_.
The scalar product of Ыvectors In question could therefore be written as
4 Pfc£(Z*V£ - Z£V*) ;
we also have
U. - P..Z* . (17)
t кг
In order to Interpret u_ geometrically, we choose rectangular axes, the first
two of which are parallel to the plane of the blvector. We will obtain
«1 --P12Z2' U2-P12Z1' V °.··Λ-°
The following properties are easily seen:
1 The vector u_ is zero when the vector z_ ie perpendicu lar to the plane of
the bivector;
О
2 When the vector z_ te parallel to the plane of the bivector (and then we
may assume It to be parallel to Ox,), the vector u_ ie obtained by
rotating the vector £ through 90 parallel to the plane of the bivector in ite
poeitive вепае, and by multiplying by the magnitude of thie bivector*
3 In the general case, the scalar product ie the вате if the vector z_ were
replaced by ite orthogonal projection on the plane of the bivector.

10
Cartesian Co-ordinatee
In 3-dlmenslonal space, the Inner product of а Ыvector a_ and a vector
z. Is equal In magnitude to the product of the Magnitude of the Ыvector a_,
the length of the vector z^ and the cosine of the angle of the vector with
the plane of the Μ vector. In the general case, we denote by Φ, the angle
Indicating the direction of a vector z. with the plane of the Mvector a_,
such that
Cos φ
111 · IH
11. The set consisting of any number of Μ vectors Is known as а ау в tan
of bivectore. The scalar product of a system of Mvectors and a given Ыvector
Is defined as the algebraic sum of the scalar products of the Ыvectors of the
system and the given Μ vector.
Two systems of Ыvectors are said to be equal If they have the s«k scalar
product with an arbitrary Ыvector. It follows that a system of Directors, If
It Is not regarded as two distinct systems of equal Ыvectors, Is completely
defined by the W'T ' quantities obtained by adding the components with the
same indices of the bivectors of the system. A system of Ыvectors has some
contravarlant components PtJ and some соvariant components P.»
As we had done for а Ыvector, we may define the scalar product of two
systems of Ыvectors, the Inner product of a system of Ыvectors and a vector,
etc.
We often assign the name biveator to every system of Ыvectors, strictly
speaking the Ыvectors are then said to be single bivectore.
In the case of three dimensions, every blvector Is simple. Effectively
let Ρ , Ρ1, Ρ be three arbitrary numbers. Let us take two Independent
solutions:
In one case X1, X2, X3, and In the other case Υ1, Υ2, Υ3,
of the linear equation
ρ23χ1 + ρ31χ2 + ρ12χ3 я 0
From this we can straight awey deduce that
X2Y3 - Y2X3 _ xV - ΧΊΥ3 _ X]Y2 - X2Y]
p23 p31 p12
It thus suffices to multiply the Y1 by a conveniently chosen factor ρ so
that the simple blvector defined by the vectors £ and p£ has the given
components.
In four dimensional space, this theorem Is no longer tenable. Besides,
It can be quite easily proved that a simple blvector with components Pv,

Cartesian Co-ordinatee
11
In a space of any number of dimensions. Is characterised by all relations of
the form
pbjpkl + pikplj + pilpjk я q
Every Mvector Is evidently equal to a system of Я'Т ' simple Ыvectors
respectively parallel to the planes formed by two co-ordinate axes. It
suffices then, to choose the Μ vectors whose components are all zero save one
pij β
12. The construction of a simple Ыvector starting with two given vectors»
may be regarded as a kind of multiplication. If we agree to denote by [x_ yj,
the Ыvector formed by the two vectors x_ and y_, we can easily verify the
relations
[x yj « -[y_x]
and
[(x + yJll ■ [xz] + Czi] *.
the multiplication Is dietHbutiv* but non-coMnutatlve, and the product changes
sign with the order of the factors.
The Ыvector [x_ yj Is known as the extevloT produot of the two vectors
x_ and y_.
If χ - Хг'ег у_ - Y£ei
are given, then by applying the rules of exterior multiplication, we obtain
[x yj - χ£γί'[%%] ■ * (XV - зеУ) [e^.] - Η P^C^.%]
which In turn gives the decomposition of a Mvector Into w*wa ' simple
Ыvectors provided by the w'wa* ' co-ordinate planes.
III. TRIVECTORS
13. A simple triveotor Is the configuration formed by three vectors
arranged In a certain order. If three vectors OA, OB, ОС, equipollent to the
three given vectors, are taken through a point 0, then the resulting three
dimensional hyperplane, (or tHplane) determined by the points О, А, В, С Is
known as the triplane of the tr1vector; Its direction alone Is defined In the
sense that It can be displaced parallel to Itself.
Two simple tr1vectors will be said to be equal If their triplanes are
parallel, the parallelplpeds constructed by three vectors taken over the same

12
Cartesian Co-ordinatee
point contain an equal volume and If the orientation of these two parallelpl-
peds Is the same.
In three dimensional space, the volume V of the parallelplped
constructed by the three vectors £, y_, z_ Is given by the determinant
,123
of the components of these three vectors.
123
Effectively the ratio Ρ does not change when we add to the vector z.
a sun of multiples of the vectors ж and y_. The volume V does not change,
since the base (constructed by £ and y) and the height do not change.
Neither does the determinant, by adding to the third line multiples of the
pi 23
first and second lines. With changing ^-n— we can then go back to the case
where the vector z_ Is taken along 0x3. Similarly on taking χ And y_
along Oxj, Ox» respectively the following formula Is obtained;
,123
xVz3
~V v~
where the right hand side Is easily seen to equal to the volume V
parallelplped constructed on the three unit co-ordinate vectors.
In order to calculate this volume V we recall that In recti
co-ordinates the volume of a given parallelplped Is given by
of the
and on squaring
V2·
we obtain
X2. X.Y.
t t t
Y.X. Y?
ъ г г
Ζ.Χ. Ζ.Υ.
г г г г
Χ.Ζ.
t t
\h
ζ2
г
x_
2 · X
i
ι-χ.
Л
On applying this result to the parallelplped constructed by the three base
vectors e_,» e-, e_3 we obtain

Carteeian Co-ordinatee
13
'Π
'12
'13
921 922 923 = 9
931 932 933
Consequently, the volume V of a simple trivector 1s given to be
(18)
As for the covariant component
V - V · P123 - ^ ■ P123
о "
123
1
Z2 Z3
it can easily be seen that by replacing the Χ., Υ., Ζ. by their values and
XXX
proceding as in No. 18, we obtain
',.з-'·'"3
then in particular
(19) V2 =· Р12зр123 ·
14. In a space that is greater than three dimensions, we can prove as we
had done for the case of the bivectors, that the condition for equality of two
simple trivectors is the equality of the contravariant components of these two
trivectors
ptj'fc я
X*7
yd
or of their covariant components
X,
ijk
X.
t
Y.
t
z.
t
X.
J
Y.
J
z.
The square of the magnitude π of a trivector is
m23 1 p. ptf*
6 tj«

14
Cartesian Co-ordinatee
We shall define the scalar product of two trlvectors In terms of one or other
of the two equivalent expressions
1 ρ ntf * = - PW0
6 vfc4 6 Y 4jk ·
that we shall Interpret as we did for the Ыvectors.
We can finally define the general trtvectore and prove that the
formulation of a simple trivector could be regarded as an exterior multiplication.
IV. MULTIVECTORS
15. Generalising the preceding Ideas naturally comes about once we
define the volume of an n-dlmenslonal parallelplped constructed by η given
vectors. In the case of n-dlmenslons (n>3). This volume In rectangular
coordinates will be defined by the determinant of the projections of the η
vectors. In Cartesian co-ordinates the volume determined by the η unit
co-ordinate vectors is equal to the square root of the determinant g of the
quantities g...
If ρ Is an Integer (p й η), a simple p-vector Is defined as the set
of ρ vectors arranged In a certain order. Two ρ-vectors are said to be equal
If they have the same contravarlant components.
Тг2
г1 l2
X ' X &
Υ ' Υ c
U ' U *
t
γΡ
ιΛ>
or the same covarlant components
t-l (Ϊ-Λ «... «t-
X. X.
г1 г2
г1 г2
г1 г2
where £,y_,...*u_ denote the p vectors defining the p-vector In question.
The square of the magnitude of a p-vector Is

Cartesian Co-ordinates
15
I p. ρ ρ
pi τ1,τ2 гр
The scalar product of two p-vectors, the systems of two p-vectors may be
defined accordingly.
If p*nt the magnitude of an η-vector Is
r-p12...nM lp B /pl2...n ρ
* P ГР12...« /P P12...n
V. SUPPLEMENTARY MULTIVECTORS
16. We shall call a supplementary multivector of a given simple p-vector
a_» the (n-p)-slmple vector b which satisfies the three following conditions
о
1 The (n-p)-plane of Ь га completely normal to the p-plane of a, that
is to ваул each of the p-vectors which define a_ is perpendicular to
eaoh of the (n-p) vectors which define t>.
2 The multiveotore a_ and t> have the вате magnitude*
3 The η-vector formed by the p-vectors of a_ and the (n-p) vectors of Ь
have a direct orientation.
This last condition assumes an oriented space, that Is to say, It was
decided a priori that such an η-vector was positive or negative, we shall
assume that the unit co-ordinate vectors are chosen In a way that a positive
n-vector Is defined by taking the natural order of indices.
Let us first of all consider the most simple case when η ■ 3, and let
us proceed to determine the vector г_ supplementary to a given Ыvector
Firstly, the following equations are obtained
ZjY1 + Z2Y2 + Z3Y3 - 0
ZjX1 + Z2X2 + Z3X3 - 0
from which
Zl Z2 . Z3 . .
-p -pi JH λ
a similar calculation would yield
ιλ-ί -J .v
23 r31 r12

16
Cartesian Co-ordinatee
To determine λ and y, let us start from the equality
Z.Z »ΗλμΡ..Ρ
г tj
Μ
from which λμ - 1, by the second condition of definition.
Thus the trlvector [x_ y_ zj has for Its contravarlant components
J
Z2 Z3
zlp23 + z2p31 + z3p12 , ,^ρ^ρν
The magnitude of this trlvector,
Is equal to the square of the magnitude of the Μ vector. It can then be seen
that
1
a
\ - /5 ·
The eupplementary vector of the given biveator therefore hae for its σοπφο-
nente
(20)
^P
23
z2 - ^p
31
ZMp ZMp
Г9 23 ' Г9 31
z3-tfP
^
12
12
This vector Is more commonly referred to as the vectorial product of two
vectors £ and y_.
Conversely, the Ыvector [x. iJ Is supplementary to the vector z.
17. The Inner product ν of the blvector [x./] *nd a vector ^j, has
been defined In (10) by the relations
v, - p21 u2 ♦ p31 u3
v2 - p12 u1 ♦ p32 u3
V3 = P13 Ul + P23 "3
By Introducing the supplementary vector z_ of [xy], wa obtain

Cartesian Co-ordinates 17
Vj - /g (Z2U3-Z3U2) , V2 - /g (zV-zV) ,
v3 - /5 (zV-zV) .
The vector ^ Is therefore the supplementary vector of the Ыvector
defined by the vector ζ and the vector u. With the ordinary vectorial
notations we have
Lx /] " £ ■ (х.л£) Λ ϋ
It Is quite easy to verify that If а Ыvector Is regarded as a geometric
sum of several others, then the supplementary vector of the given Ыvector Is
the geometric sum of the supplementary vectors of the component Ыvectors.
18. In the general case, the components Q of the {n-p) supplementary
vectors of a given p-vector, with components P, are
JTJ2 Jn~p
(21)
J λ*3'г Jn-p ι
n-p ш Ι ρ
/5 г'т£2 ip
It Is assumed in these relations that the Indices t.,t-,...»t ./.„/„...J
\ с ρ ι с n-p
are, to the nearest order, all of the indices 1,2,.. .,n, and that the
resulting permutation Is even.
It can be seen that If b. Is the supplementary multlvector of the
p-vector a_, the ρ vector a_' supplementary to b is a or -£; the
last case only occurring If ρ and {n-p) are odd (and consequently, If η
Is even).
The supplementary rnultiveotor of a given non-simple ρ vector Is defined
as the set of the supplementary multlvectors of the simple p-vectors of which
the given p-vector Is the sum.
In the case where n-4 and ps2, the supplementary Ыvector of a
Ыvector with components PtJ or P.., has for Its components
Q23 - J5 P14 . Q31 - Ъ P21 . 012 - fi P34
o,4 - ^ p23 . o2, · я p31 , o34 - я p12
or
023 . _L ρ η31 - J- Ρ О12 - J- Ρ
o14 . -L ρ η24 - -L ρ ο34 - -L ρ
Q $ P23 · Q $ P31 · Q $ P12

18
Cartesian Co-ordinates
We ascertain that the scalar product of а Ыvector and Its supplement Is
zero when this Ыvector Is simple, and vice-versa.
VI. SLIDING OR BOUND HULTIVECTORS
In the sime way that free vectors are distinguished from eliding vectors,
free «ult1 vectors may be distinguished from eliding multlvectors. A simple
sliding p-vector Is formed by ρ vectors situated in the вате p-ptanef and
equality only holds when they are situated In tie same p-plane. The
components of a free ρ-vector previously dealt with do not suffice to determine
a sliding p-vector.
Let us take a point χ ....χ" of the p-pline containing the p-vector.
To fix Ideas, let us take p-3. The equations of the trip lane which contain
a tr1 vector [x^z] are obtained by equating to zero all determinants of
order 5 of the matrix
Xй 1
x"o
Yn О
ζ*ο
.1 Л
where ς ,ς »..-,ζ" denote the current co-ordinates. It can be seen that
the quantities appearing In these equations will be· via the Ρ
quantities
x* tf> x* >
X* X^ X* ]
ijk
the
№1
γϊ yj уЛ
z* r> z*
The Ρ
ijk
can themselves be expressed In the form
Y°.Z°
tpijk
by agreeing to have
χ - 1 and Xv - Yv ■ V - 0. The sliding trlvector can then be regarded as
an exterior product [H ж ^ zj where Η denotes a point, however arbitrary,
In the triplane of the trlvector. The rule of formation of the components
Is the same as for a free 4-vector, by assuming It to be In (n+l)-d1mens1onal
space, the (n+l)-th co-ordinate of Η being 1 and the (n+l)-th co-ordinates
of the vectors x_, £, z_ being zero. If p* 1 and n-3, we recover the six

Cartesian Co-ordinateв
19
classical co-ordinates of a sliding vector. A system of eliding p-veotore 1s
similarly defined.
VII. APPLICATION TO THE MOTION OF A RIGID В00Υ HAVING A FIXED POINT
20. When a rigid body is set 1n continuous motion, the velocity field
has a fixed point that can be assumed to be the origin of the co-ordinates, and
at Its different points, 1t has the fundamental property that the velocities
of two arbitrary points Η and H' have the same projection on the Hne MM1.
If x1 and у denote the co-ordinates of Η and H' respectively and
u. and v. denote the covarlant components of their respective velocities,
then we have
{v. -«.)(/- x£) - 0 .
X- X-
On the other hand, the velocity of Μ being normal to OM, and that of
H' normal to OM1, we have
t t n
u.x ■ v.у ■ 0
whence the relation
■£ i «
u.y + v.χ * 0
By taking M1 as the point situated at the extremity of the vector е.,
this relation shows that u. 1s a linear combination of constant coefficients
к г
of the co-ordinates χ,
4 ' «Η** (22)
and the perpendicularity of the velocity u, with the straight Hne OM gives
t к i n
The quantities a., are thue anti-eyrrwnetrio (a*. - -a· J; they define а
Ы vector (*): the velocity of a point Μ ie the inner product of the biveo-
tor with oontravariant components a. . and the vector OM.
We shall say that the body 1s subjected to an Instantaneous rotation
represented by the Ыvector with components a...
In the same way
i ki ,
и - a xk , (23)
(*) See No. 23, footnote (1).

20
Cartesian Co-ordinate в
will be obtained by Introducing the contravarfant components of the Ыvector.
The rotation 1s said to be simple when the Ыvector 1s eimpZe; 1n this case
the points which lie 1n the (n-2) plane being the locus of the normals erected
at 0, to the plane of the simple Ыvector, all have zero velocity. The
rotation is described about this (n-2)-plane. The angular velocity Of rotation 1s
moreover, equal to the magnitude of the Ыvector.
For the case n-3, all rotations are simple. The most general rotation
1s decomposable Into three rotations about the normals to the three co-ordinate
planes, their respective angular velocities being
23 / Г 31 / Г 12 / Г
a *дг$гг~дгг · a *Ч$\\-дг\ · a /дл\дгг'д\г -
If the vector с 1s introduced supplementary to the Ыvector a which
represents the rotation, then this could also be decomposed into three rotations
about the respective co-ordinate axes, their respective angular velocities in
this case are
,— 1 fi\\ ,— 2 fill ,— 2 /*33
^Y\° V~a23 · ^22 σ /~a31 · ^33 σ J—a\V
Generally, a rotation could also be represented by the (n-2) vector
supplementary to the Ыvector a.., as we have in ordinary space.
VIII. TENSORS, TENSOR ALGEBRA
Vectors, bi vectors, multi vectors, or rather the numerical array в wHoh
define them analytically, are particular types of tensore. We generally refer
to a tensor as such an array defining, analytically, a geometric (or physical)
object, such that by a change in Cartesian co-ordinates, the components of the
tensor undergo a linear transformation whose coefficients do not depend on
the numerical values of these components, but uniquely on the two systems of
co-ordinates. Alternatively, we say with more precision that it 1s the array
of coefficients of the transformation that carry the contravariant components
of a vector by the change of co-ordinates in question.
We shall say that the tensors whose components obey the same rule of
transformation for the same change in co-ordinates, forma kind of field; we
shall also say that they are of the same nature. It 1s important to note that
the nature of the tensor field 1s defined uniquely by the rule of
transformation of its components in a way that 1s independent of the geometrical,
mechanical or physical significance of the objects that these components represent
analytically. From this point of view, the contravariant and covariant
vector fields are to be regarded as two distinct tensor fields, since the

Carteeian Co-ordinate e
21
transformation of the contravarlant components of a vector for a given change
In co-ordinates 1s not the same as that undergone by the covarlant components.
In the contravarlant (or covarlant) vector fields, the components are
linearly Independent; that 1s to say they do not satisfy and linear and
homogeneous relation with constant coefficients· The tensors that we shall
consider will always cone from a field with linearly Independent components.
Teneor algebra provides the laws for the formation of tensors. Let us
first describe an Important theorem which permits recognition of the tensor
character of certain objects, represented analytically by a certain finite
number of components.
Theorem. Let (С) be a teneor field with r linearly independent oomponente
UijUpj...,» and on the other hand, let there he an object aueoeptible to
being represented analytically in a Carteeian ос-ordinate eyatem by r quon-
titiee ν 3v ,...,, ur, euoh that the ew\ u.v1 ie independent of the ohoioe
of oonponente, the u. being the oonponente of the arbitrary teneor field
(C). Under these oonditiona we have:
The quantities υ form с teneor
?
2 The nature of this teneor га uniquely determined by the nature of (€)
1.
The proof 1s quite straightforward. Let (u.)' " X.u, be the transformation
of the components of a generic tensor of the field {€) due to a change 1n
Cartesian co-ordinates. If the quantities which define the object 1n question
1n the new system of co-ordinates are denoted by (v *)'» then we obtain the
relation
\\uk («*)■- u/ ,
and beoauee of the linear independence of the u, , we have
vk - \\ (v£)' .
This last relation defines without ambiguity, the linear transformation which
takes the vb to (v1)'; the two parts of the theorem are thus proved.*
A particular case of this theorem leads evidently to the tensor character of
the covarlant vectors. It suffices to take as the field {€) the field of
contravarlant vectors, each of which 1s represented by η components u.,
i l
such that the sum u.X being Independent of the choice of co-ordinates, 1s
*
The two linear transformations undergone by u. and v. are said to be con
tragredlent. г

22 Cartesian Co-ordinates
a tensor and where the Хг denotes the contravariant components of a generic
vector. The tensor here 1s of the sane nature «s a covarlant vector.
22. Tensor algebra consists of several operations which govern the way
two tensors result 1n a third tensor.
The first of these operations 1s addition* which takes two tensors of a
given valence to a third of the same valence, and whose components result from
the addition of the homologous components of the two given tensors. The
geometric addition of the contravariant and that of covarlant vectors, are
particular cases of this operation.
A second operation 1s general multiplication. Given two tensors,
Whether or not of the same nature, the quantities obtained as the result
of multiplying 1n all possible manners, the components of the first tensor by
those of the tecond, constitute a third whose nature depends only on that of
the first two. If a change of co-ordinates subjects the components u.
(г - l,2,...,p) of the first tensor to the transformation
<«i)' - λΚ
and those of the second to the transformation
then the products u.\». undergo the transformation
A very simple case 1s provided by the general product of two contravariant
vectors X and V7, whose components are the products XT; the general
product of a contravariant vector Хг and a covarlant vector Y., has for
Its components, the products ХгУ ..
The preceding operation combined with the operation of addition (or
subtraction) leads to a simple contravariant Ыvector P1"*' which 1s the
difference between the tensor X Y*7, and the tensor of the same nature Υ Χ"\ By
adding simple Ыvectors, the most general Ыvector 1s then obtained. It can
therefore be seen that the components PtJ of a biveator, transform as the
о&гфопапЬа of the general produot of two oontvai&inant veotore.
23. The above considerations lead to tensors most frequently used:
these are tensors having several Indices, one type being "upper"
(contravariant Indices), the other type being "lower" (covarlant Indices). For example,
a tensor with two covarlant Indices a., may be characterised by the property
i i i i
that the awn a. .X Υ" , where X and Y" are the atmtravariant components

Cartesian Co-ordinates
23
of tuo arbitrary veators, hoe α nmerioal value independent of the ahoioe of
ao-ordinatea. *
Theorem 21 tells us 1n effect that the components a., define a tensor
t i 3
whose nature only depends on that of the tensor X r whose components are
linearly Independent, «hen the quantities a., · U{V,· where U^ and V. are
the covarlant components of two vectors satisfying the above stated condition.
the product а,ЛгТ* - U.V.XT has a numerical value Independent of the choice
ij г j
of coordinates. The result 1s that the tensor a.. 1s of the same
nature as the product of two covarlant vectors, or better still, by a change
1n co-ordinates, the components a., undergo the same linear transformation
as the products U.V..
г J U
A contravarlant tensor α y with two Indices, 1s similarly characterised
by the property that the sum at,7X.Y., where the X. and the Y. are the
covarlant components of two arbitrary vectors, has a numerical value
Independent of the choice of Cartesian co-ordinates. The components atJ transform
as the components l/v7 of the general product of two contravarlant vectors.
Finally, a mixed tensor with two Indices aJ. 1s characterised by the
property that the sum dixlY. 1s Independent of the choice of co-ordinates,
the Хг and the Y. denoting the components of two arbitrary vectors, the
first contravarlant, the second covarlant. The components cc. transform as
the components U.r of the general product of a covarlant vector and a con-
travarlant vector. Similarly tensors with three Indices are characterised
as being either contravarlant, covarlant or mixed, I.e. a** , a**w *i> etc*
It 1s necessary to assume that the order 1n each case, spans the three Indices
occurring 1n the definition of the tensor. In particular oases, the
components do not change 1n numerical value 1f, for example, the order of the
Indices are reversed. In the general case the tensor has η linearly
Independent components, but 1t could happen that these are constrained linear
relations with constant coefficients. It 1s then necessary for these coefficients
to be the same whatever the system of Cartesian co-ordinates employed. He must
bear 1n mind that 1n this case, the number of components 1s actually less than
3
η .
Let us assume for example that the tensor a... 1s anti-symmetric. This
means that the numerical values of the components a.., do not change under
an even permutation on the three Indices, and the sign changes 1f an odd
permutation 1s carried out on these Indices. It 1s Important to note that if
this property holds for a particular system of Cartesian co-ordinates, then
This 1s what happens to the a^· that represent the field of motion of a
rigid body about the origin (No. 20), for the scalar product of a vector χ
and the velocity of the extremity of a vector зс» 1s equal to aw Χ*Υ£. on
account of (22), and Its numerical value 1s Independent of the choice of
coordinates. The a^ are thus the components of a tensor having two Indices
and as these components are ant1-symmetr1c, 1t 1s a Ыvector. Thus justifies
the claim made in No. 20.

24
Cartesian Co-ordinates
it holds for all such systems. In fact the a.., transform as the products
X.Y.Z. where the components of three arbitrary vectors зс, χ, _z enter.
Consequently they transform equally as the quantities
hVk ♦ hzA ♦ W* - W* - hbh - W* ·
or as the components of a covarlant trlvector. Instead of having η linearly
«iw-lНя-21
Independent components, there are actually ^—f*2—L . An antisymmetric
tensor with three Indices 1s thus a trlvector.
In the same way a symmetric tensor such as a£ fc admits "fr1*1^"*2)
Independent components; they transform as the components X.» X.» X. of the
l· J rC
product of three Identical covarlant vectors, between themselves.
A tensor with several Indices could be taken analytically 1n several
different forms; 1t suffices to Introduce, for example, 1n the case of three
Indices, the quantities a^^fc* 3<zjf' 'a%^ def1ned b* the 1den_
tH1es
a..,, XVZ* - a1.. X.**Z* - u[J' X.Y,Zk - uiJk X.Y.Z, .
tjk jk г к г j г j к
Care must be taken when a lower Index 1s raised to the upper row, with
respect to the order of the Indices; sometimes dots are placed above the lower
Indices, and below the upper Indices, by writing a. ., Instead of a ..
An Important type of tensor 1s the fundamtntal tensor which has the д..
for Its components. It 1s a well-defined tensor, as the sum д.. X r has
a numerical value Independent of the choice of co-ordinates, since 1t 1s the
scalar product of the vectors χ and χ. Its mixed components tf. or g
are defined by the Identities
A x4- * sj xiw * χίγί ■ ν* ·
we have then
The contravariant components gtJ of the fundamental tensor are defined by
the Identity
from which
Y g Yj '
these are precisely the quantities that were previously Introduced with this
notation in No. 4. Raising an Index to the upper set 1s furthermore governed

Cartesian Co-ordinatea
25
by formulae such as the following
4-A* ■·
similarly, we have
24. From a tensor a., with two Indices, a scalar tensor could always
be deduced (that 1s to say as a constant Independent of the system of
coordinates) by an operation known as the eaturation of the indiaea. This scalar
tensor, known as the contracted tenaor of the given tensor, 1s a.. The proof
1s Immediate. The components of the mixed tensor J, transform between them-
selves as the components X.r of the general product of a covarlant vector
г i
χ and a contravarlant vector χ. Now the 1ηvariance of the sum X.Y ,
which represents the scalar product of the two vectors, Implies the tensor
property of the sum a1.. It 1s Important to note that there 1s a second con-
i l
tracted tensor a., but this 1s equal to the firat. In fact by a calculation,
we find
i „ ki i „ ik
ai- 4k* - Ч ' Чкл
The symmetrical property {д.. - д..) of the metric tensor suffices to prove
the equality of two contracted tensors. The contracted tensor of а Ыvector
1s zero, since we have
i . , ki . iks n
ai = hglk{a +a ) = ° ·
Tensors with more than two Indices also acfcnlt contracted tensors. Take
for example the tensor a..,,. We raise the last Index to obtain the mixed
tensor a..,. Its components transform as the components of the general pro-
duct Χ.Υ.Ζ,ΙΓ of four tensors x,y, z, u, 1n which the first three are covar-
iant and the fourth contravarlant. Now the sum Χ.Υ.Ζ,ΙΓ, being the product
l· J rC
of X.Y. and the scalar product of the vectors ζ and u, transforms as
X^Y.; consequently it is the same as the quantity
г j tjk
The b.. are thus the components of a covarlant tensor with two Indices; 1t
1s the given tensor contracted by aaturation of the laat too indiaea.
Other cases may be considered by saturating two Indices 1n different rows.
If ρ s4 every contracted tensor gives rise in turn to other contracted
tensors, and so on.
In the case of a raultivector, the contracted tensors are all zero.

26
Cartesian Co-ordinates
25. A last operation 1s frequently employed, it 1s oontraoted
nmltriplication. This consists in once or twice contracting the general product of two
given tensors. The products of the fundamental tensor and an arbitrary tensor
aiik* once contracted* take the form
*£ejV *iaik>·"' 9iajk> ""
The numerical values of the nixed components of the metric tensor being given,
all these contracted products yield the tensor a-.. Itself.
The fundamental tensor in these operations thus plays the port of unity.
One of the contracted products of а Ыvector a.„ and a vector Γ,
к l*
is α,Λ ; it is precisely the Inner product of а Ыvector and a vector (see
No. 10).
Ноге generally we can define the contracted product of a p-vector
a. . . with a ^-vector b. . (я*р) as the product q times
contracted^ ' £ *
С р - α ίί*1 Ρ Ь
.Г2...^
or rather as its quotient by q\
This Inner product would permit us to define and to form, the cosine of
the angle between a p-plane and a q-plane.
26. We shall complete this section by applying the preceding theorems
to show how the components of the supplementary multlvector of α given nultl-
vector may be determined.
Firstly, let us note the tensor character of the square root of the
discriminant of the fundamental form, i.e /g. In n-di mens ions then-vector with
12 и
unique contravariant component ρ 1s a tensor (but it is not a scalar
tensor, as Its unique component 1s a variable with respect to the system of
co-ordinates). On the other hand the fact that Jg p12···", the volume of
the n-vector, 1s Independent of the choice of co-ordinates, proves by virtue
of the theorem in 21, the tensor character of Jgt and simultaneously shows
that /g transforms as a covarlant n-vector.
The result of this 1s that given a contravariant Ыvector with
components ?tJ, the qinntltltes Jg P1,7 ere the components of α tensor. To fix
Ideas, let us assume that n-5. Following the above discussion, the
quantities Jg PtJ transform as the quantities Qi23arp on denoting by Q12345
the component of a covarlant 5-vector; but Qi9345p for example is one of
the components of the contracted product Q--;y.i^Z of the covarlant 5-vector
and of the Ыvector. Now, these components transform as those of a covarlant
tr1 vector. Consequently the tensor Jg P^ has the sane nature ae a oovari-
ant triveator, with the component Jg Рг^ corresponding to the component

Cartesian Co-ordinatee
27
Q..- of the tr1vector, such that the permutation [ijkhl) 1s even. Thus a
determined tr1vector 1s associated to the Ыvector PtJ.
To Interpret this trlvector geometrically, for the case where the blvec-
tor Ρ ^ 1s simple, let us choose rectangular co-ordinates with five basis
vectors e. of which the first two, e, and e^, are taken 1n the plane of
the Ыvector; this Ыvector will then have a single non-zero component, I.e.
12
Ρ , equal to the magnitude m of the Ыvector. The desired trlvector will
consequently have a single non-zero component, Q34C» equal to m; 1t 1s thus
a simple trlvector whose trlplane 1s entirely normal to the biplane of the
Ыvector and has an equal magnitude.
He may refer to the definition of the supplementary trlvector of the
Ыvector as 1t had been given 1n section V
The tensor Jq Ρν provides us with an example of a more general type
than that discussed 1n Nos. 21 to 25.

Chapter II
CURVILINEAR COORDINATES IN EUCLIDEAN GEOMETRY
I. THE SPATIAL LINE ELEMENT IN CARTESIAN COORDINATES.
27. He have seen that the square of the length of a vector 1n any system
of Cartesian coordinates 1s expressed by means of a positive definite
quadratic form 1n the contravarlant components of the vector, viz:
9,j X1XJ · (1)
Conversely, given a priori a positive definite quadratic form 1n n-variables
X ,...,Xn, 1n a given system of Cartesian coordinates, then this form defines
the square of the length of a vector whose components with respect to this
system will be X ,...,X .
When such an expression as (1) 1s positive definite, then 1t may be
reduced to a sum of η Independent squared terms, I.e. there are η linear
forms Y] Yn of the X1 satisfying the Identity
Yl + Y2+ ·'· + Yn = 9ij Χ^ ' <2>
This being the case, let us take an arbitrary system of rectangular
coordinates and consider the vector χ whose components with respect to this system
are Y,,Y2,...,Y . By hypothesis, a relationship of the form
Y1 ■ «Ik Χ"
1s obtained; we shall consider the vectors e,,e_2,...,e whose projections
on the axes, are for a particular е.,
α11'α21'··"αη1
If then a system of Cartesian coordinates 1s considered for which the η
basis vectors are precisely e,,...,» , 1t is straight away seen that the
contravarlant components of the vector x, 1n this system of coordinates,
are X, X ,..., Xn. The Identity (2) shows then that the given quadratic form
properly represents the square of the length of the vector χ 1 η this system
of Cartesian coordinates. As the system of rectangular coordinates originally
Chosen 1s arbitrary, it can he seen that the ау а tern of Cartesian coordinates
corresponding to the given quadratic form га well defined up to a displacement
or up to a displacement and symmetry.
29

30
Curvilinear Coordinate
In order that a quadratic differential form g. du a\f with aoratant
coefficients can be regarded as the square of the distance between tuo points
infiniteeimally close in a system of Cartesian coordinates, it is necessary
and sufficient that this form should be positive definite,
28. When oblique coordinates are considered 1n ordinary analytical
geometry, a unit vector whose magnitude 1s one unit of length, 1s usually taken on
each of the axes. The quadratic form which gives the square of the magnitude
of a vector (Χ,Υ,Ζ) 1s then
χ2 + γ2 + Z2 + 2 Cos λ YZ + 2 Cos у ZX + 2 Cos υ ΧΥ , (3)
where λ, у, ν denote the angles between the faces of the trihedral of the
coordinates.
The neoessary and sufficient condition for three anglee λ,μ,ν taken
between 0 and тг, regarded as the faces of a trihedral, is that the
quadratic form in (3) should be positive definite.
The decomposition Into squared terms gives
(X ♦ Υ Cos ν ♦ 2 Cos y)2 ♦ (Y S1n ν ♦ 2 Cos λ iff* Η Cos u)2
+ Sin2 у Sin2 ν - (Cos λ - Cos у Cos v)2 Z2
S1n2 υ
The required condition 1s thus
(Cos λ - Cos у Cos v) - Sin у Sin ν < 0
or alternatively
|Cos(y + v) - Cos λ| |Cos λ - Cos (y-v)| > 0
If we assume that λ 1s the largest of the given angles, the second
factor on the right hand side of this Inequality 1s Itself negative, and the
condition required 1s
Cos λ > Cos (y+ v)
or, which amounts to the same thing,
λ<ν + υ<2ττ-λ

Curvilinear Coordinates
31
He recover the classical conditions that each face 1s smaller than the
sun of the other two and that the sun of the faces 1s less than four right
angles. It can be seen that these conditions are sufficient.
II. THE FUNDAMENTAL THEOREM OF METRIC GEOMETRY
27. The two main problems posed by the theory of curvilinear coordinates
1n Euclidean geometry, are the following:
1. Having been given the spatial line element in a ayetem of curvilinear
coordinate*, to determine the nature of these coordinates, or what amounts to
the same thing, to pass from these coordinates to rectangular coordinates.
2. What oondttions must be satisfied for the coefficients of an a priori given
line element, for this line element to be regarded as that of Euclidean space
in a system of curvilinear coordinates conveniently chosen?
*
These two problems had been first dealt with by Lame, for the case of
orthogonal coordinate systems. Having concerned outselves with the first of them 1t
1s essential to take account a priori of the number of possible solutions 1t
has. Let
ds - g1 .du1duJ (4)
be the spatial Hne element 1n an arbitrary system of coordinates. At each
point M, with coordinates u , u ,...,un, 1t 1s possible to attach a
Cartesian coordinate system 1n which Μ will be the origin and whose basis
vectors e,.e,,....e will be chosen such that the coordinates of the point
M'(u]+ du!...,un + dun)
1nf1n1tes1mally close to M, are precisely du ,...»dun. For this 1t
suffices to have the vector e. tangent to the 1-th coordinate curve (obtained
1n varying the single coordinate u ) and 1n a more precise way, to represent
the velocity of a point travelling along this curve with the variable
coordinate u taken as time.
In this system of Cartesian coordinates which defines what we shall call
the natural frame associated with the point M, the scalar product of the two
vectors *f and e_. 1s g^., and consequently the angle Φ^ between the
1-th and the j-th coordinate lines 1s given by
911
(5) Cos φ< υ
PU
ЭД7
*
G. Lame, Leoons sur les oo-ordonnies ourvillignes; Paris. 1859.

32
Curvilinear Coordinates
In a more general way, the cosine of the angle φ between two directions
steuning from Η and defined by the two symbols of differentiation d and 6,
1s
gH du 6ir
Cos φ - bi
/g^ duW /g1. ouW
30. With this established, we now assume that 1n ordinary 3-d1mens1onal
space, the line element takes the form
2 2 2 2
ds - du + d/ + dw (6)
It 1s clear that this is the case 1n any system of rectangular Cartesian
coordinates. We are going to show that there does not exist any other system of
coordinates giving the line element the form in (6).
Effectively, 1f the straight lines of the space are to be determined,
their characteristic property being that they are the shortest path from one
point to another, then it 1s necessary to minimise the Integral
\A
'du' + dvZ + dwZ
When u, v, w denote the rectangular coordinates, the analytical problem
that results leads to lines defined by a system of two first order equations
1n u, v, w. But the result of the calculation, depending only on the given
expression /lu2 + dvz + dw2, 1s general and the straight lines are defined
by the linear equations, whatever the nature of the coordinates u, ν, ω. The
coordinate curves are thus all straight lines, respectively rectangular between
themselves on account of formula (5). The eurfaoea vith coordinates и - const.
are planes, since every straight line which has two of Its points 1n one of
them continues Indefinitely. The straight lines with coordinates (w), all
being perpendicular to the plane w»0, are parallel to each other. It 1s
easily deduced that u, v, w, are the distances from a point 1n space to the
three fixed rectangular planes u-0, v-0, w-0. Consequently, the
coordinates 1n auest1on are necessarily Cartesian and rectangular.
This argument may be presented under another guise, which has the
advantage of being applied to any form of line element uhateoeoer.
Let us take two coordinate systems, that may or may not be curvilinear,
giving the ordinary spatial Hne element 1n the same form:
ds2 * g1 .du1duJ
Let Η and H' be two points 1n space which admit 1n these two systems,
12 3
the same given coordinates u , u , u . The correspondence so obtained between

Curvilinear Coordinates
33
the points defines a point transformation (which enables a passage from Η to
H') having the property that the distance between any two points
Infinitesimal ly close 1n the space does not change under this transformation. (A
transformation of this kind 1s said to be isometric, ) The transformation 1n
question takes one arc of a curve Into another which evidently has the вате arc-
length (since 1t 1s given by the same Integral). Consequently the distance,
between two arbitrary points, by Its very nature, 1s Invariant under
the transformation. Now 1n elementary geometry 1t can be proved that two
figures corresponding point by point with distances preserved are equal or
symmetric. Consequently the trans formation taking Η to H' is a
displacement followed by a symmetry.
If we Imagine the system determined by the three families of coordinate
lines, each of these lines being numbered (by the numerical values of the two
non-variable coordinates on that line), we can see that the too systems
attached to the too coordinate systems providing the same ds are deduced,
one from the other, by a displacement followed (or not followed) by a spmetry.
This can be restated by saying that whatever the system of coordinates provld-
2
ing a given ds , the natural frames attached to the different points of the
space (as they had been defined 1n No. 29) are always 1n the same position,
one with respect to each other. The frame attached to a system of fixed
values u «a being fixed, all the other frames are defined in that вате
way. The preceding theorem shows that all the geometric properties of the
space are virtually contained in its line element (yet with a restriction
concerning the orientation of the space). This constitutes the fundamental
theorem of metric geometry,
31. It 1s Interesting to note that the only hypotheses made 1n the
preceding arguments Involving the coordinates u are those assuring that the
square of the distance between two points inf1n1tes1mally close 1s a
quadratic differential form with respect to the du . It is insufficient 1n this
case to assume that the coordinates of two points 1nf1n1tes1mally close are
themselves 1nf1n1tes1mally close. To make up for this, 1t suffices to assume
that the u are functions of rectangular coordinates admitting continuous
first order partial derivatives. The same property will hold true for
rectangular coordinates, considered as functions of the u .
The above proofs had been made 1n 3-d1mens1onal space. The actual
solution of the first problem stated 1n No. 29, will show that the fundamental
theorem of metric geometry 1s true whatever the number of dimensions of the
space.

34
Curvilinear Coordinates
III. THE LOCAL RECONSTRUCTION OF THE SPACE IN TERMS OF ITS LINE ELEMENT
32. Let us now approach the first fundamental problem stated 1n No. 29.
Knowing the line element
ds2 - gjjduW (4)
means that at each point M(u ,...,un) a natural frcene (R) of given span
and form (to within a symmetry) can be Imagined, with this point as Its origin.
Firstly, let us localise with respect to the natural frame (R) relative to
M, the natural frame (R*) relative to a point M' Infinitesimally close to
M. The coordinate vectors e^...,· of the frame (R) are none other than
the vectors
ЭМ ЭИ
η ·
Эй Эй
expressed by the equality
dM - du1^ + du2eg + ... + du"^ (7)
On the other hand, the assumptions made about the coordinates u show that
the vectors e_j,...,e^ attached to M' differ from the β^,.-.,β^ by an
Infinitesimal amount. In order to define them analytically, we must assume
that the coordinates u , considered as functions of rectangular coordinates,
admit continuous aeoond order partial derivatives.
The direction parameters, that 1s to say the contravariant components of
the vector ej, with respect to the frame (R), are therefore of the form
.2 1-1 -. 1 η
L
where the αή are linear expressions with respect to the differentials
du1 dun .
ωϊ " Γϊΐ dyl + Γ?2 d"2 + "* + Γϊη du" (8>
and we could write 1n a more general way
3 к
The first task 1s to determine the η quantities Γ1ρ. This having been
done, each frame (R*) Infinitesimally close to a given frame (R) will be
localised with respect to (R): the Euclidean epaae \sill be reconstructed in
the neighbourhood of the origin Μ of the from (R).

Curvilinear Coordinates
35
33. Having solved the problem 1n question, let us Introduce some new
notation. Let us note that the ω?, where the Index 1 stays fixed and the
Index к varies are the contravarlant components of the vector de. with
respect to (R). Let us Introduce the aovariant components
U1J " 9Jk ωϊ <10>
of this vector and put
u1J " r1jr ^
which amounts to
r1jr -ЪкТЪ ■ (11)
These notations being clearly understood, we find that we have the first group
of relations to determine the coefficients of the expressions tal. They are
those that express how the natural frame always has the span and form Imposed
by the spatial Hne elament, I.e.
Differentiating these relations yields
91ku1 +g1ku1 " dg1j
or by (10)
"1J +UJ1 " d9U * (I)
2
The relations 1n (I) are 1n number: they are not therefore
sufficient to determine the unknown coefficients.
We shall obtain some other relations by expressing the integrobility
conditions of the equations (7) and (9)
dM - du1 e1 (7)
For the moment let us content ourselves with expressing the 1ntegrab1l1ty
conditions of (7) by expressing the vector
э2н

Эб
Curvilinear Coordinates
in two different ways; we obtain
32H _ Э ЭН ш Ъ^\ ш гк β
Эй 3uJ 3uJ Эй 3uJ 1J ^
Э2Н ж Э ЭН _ ^j . Jc .
\ j —г ~т —ι г11 \
Эй W Эй 3uJ Эй J
The relations giving the integrateHty conditions are
ru " rji (ii)
or by (II)
rikj * rjki · -(π·)
Let us put aside for the moment the 1ntegrab1l1ty conditions of the equa-
2
tlons in (9) and note that the equations 1n (II) are n ίη"Ί) 1η number; this
2
number added to the number of equations 1n (I), gives of course η ,
the number of unknowns.
To solve equations (I) and (II1)» we start from the equation
3g1j
г + г ■ —-J-
'ljk+lJ1k 3uk ·
deduced from (I), and replace Г... by Г. .. 1n accordance with (II),
э9ц
rk1j + r1jk"-^ '
A cyclic permutation, carried out twice 1n succession on the letters 1, j. к
yields two new equations
Mjk rjki 8ui
Э9к1
rjki + гки - -^ ■
It can easily be deduced that
г я ι Э91к + Э91к *ц . Г11 kl
г1к1 7 Г —J Τ " L1J'kJ
Jkl l 3u 3uJ 3u

Curvilinear Coordinates
37
*
on Introducing a notation due to E. 8. ChHstoffel [the Christoffel symbol of
the firet kind).
Conversely. 1t can be verified without difficulty that the values so
obtained for the quantities Г.. . will satisfy equations (I) and (ΙΓ);
besides the equations 1n (ΙΓ) result straight away from the symmetry with
respect to the upper Indices of the ChHstoffel symbols. Finally, the required
relationship 1s obtained,
l 3gu 33<ь эдн
г1у"гЛк1-1^к1-?тт*тт-т1[ · (12)
He now change from the Г... to Г.. by the Inversion of the relations 1n
(II), which gives
ГЪ ■ Γ5ΐ B AkJ " 94[«.« - <TJ} (13)
к
The quantities {γτ} are the Chriatoffel symbols of the second kind.
The relationships (12) or (13) solve the proposed problem completely. It
may be verified that the solution 1s unique, this being 1n harmony with the
fundamental theorem of metric geometry.
Having locally reconstructed the space from its Hne element,
we can see that by Integration 1t 1s possible to completely localise, each
with respect to the other, the natural frames attached to the different points
in space. He shall touch upon this again.
IV. ABSOLUTE DIFFERENTIATION, KINEHATICAL APPLICATIONS. LAGRANGE'S EQUATIONS.
34. Let us consider a vector field 1n the space. At each point H, the
vector of the field, with respect to the natural frame attached to this point,
has given contravarlant components X , УГ X . Let us determine the
elementary geometric variation x' - χ of the vector when the point Η passes
to a point M' 1nf1n1tes1mally close to H. On taking into account the terms
1n (9), the equality χ ■ X e^ gives
d£ " d*1^ + χ1ω^ - (dX1 + Χ1^)^ .
It can be seen that the elementary geometric variation of the vector, or
Its absolute differential, has for Its components with respect to the frame
attached to H,
*
E. B. ChHstoffel· Uber die Transformation der Homogenen Differential·
auadruoke Zveiten Gradea. J. de Crelle, t. 70, 1869, pp. 48-49.

38 Curvilinear Coordinates
DX1 - dX1 + ХкШ|[ . (14)
The quantities DX define the absolute differential of the vector of the
given field; they are themselves contravarlant components. In particular, 1f
the vector field is uniform, the absolute differential Is zero and we have
dX1 ♦ XkU|[ - 0 . (15)
It could be said that 1f a vector 1s transported from a point Η to a point
1nf1n1tes1mally near 1n such a way that 1t remains equipollent (transport by
equlpollence), then the relations 1n (15) are obtained.
It 1s Important to know how to calculate the cowzriant components of the
absolute differential of a vector with covarlant components X.. For this
purpose let us Introduce an arbitrary, uniform, but fixed vector field, with
contravarlant components Υ . The elementary variation of the scalar product
could be obtained 1n two ways. In the first place 1t 1s equal to the scalar
product of the elementary variation of the vector x. and the fixed vector £
which gives
DXjY1
In the second place, a direct calculation can be made, which gives
dXjT1 + XjdY1 .
If the equations, similar to those of (15), which the uniform field £
satisfies are taken Into account, then the second sum may be written as
1Y uk " V ω1
and finally, we have
DXjY1 - (dX1 - Xku)j[)Y1 .
Since this relation 1s true whatever the uniform field £, the sought-after
relationship 1s obtained
DX1 - dX1 - XkuJ . (16)
35. These matters may be seen from a more general point of view. Let us
attach to each point Η of the space a given point Ρ whose assigned
coordinates are χ ,...,xn with respect to the natural frame (R) relative to H.

Curvilinear Coordinates
39
He thus define a field of pointe. Let us determine the elementary absolute
displacement of the point Ρ when we pass from Η to H1. We can then write
P-H ■ x1^
from which
d(P-M) - dx1ej + xkmje. ;
consequently, on taking Into account (7),
dP « |dx1 + du1 + x^le^
By analogy with our considerations for vector fields, we shall therefore
establish
Dx1 - dx1 + du1 + xku>l , (17)
a relationship defining the absolute differential of a field of points.
36. The above results (Nos. 33 and 34) readily permit us to determine the
velocity and acceleration of a point Η 1η motion. Let us suppose that the
curvilinear coordinates u of this point are given functions of the time t.
The velocity of the point will evidently have for Its contravarlant components»
with respect to the natural frame attached to this point
v1'# ■ 08)
The acceleration 1s the derivative with respect to t of the absolute
differential of the velocity vector.
1 -1 . 1 л r1 duk duh MOi
Y ■ * ■du + rkn-ar-oT · (19>
He shall note that as the functions u of t are continuous, 1t suffices to
know at Η the numerical values of the Chrlstoffel symbols of the second kind
In order to know the contravarlant components of the acceleration. The
relation in (19) generalises the theorem of the composition of accelerations.
In a similar way the accelerations of different orders may be determined.
37. The formula 1n (19) Immediately gives us the means of finding the
equations of the straight lines 1n the system of coordinates under
consideration. It suffices to Integrate the second order differential equations
d2u1 . r1 duk duh ш n
-^?+ rkh if -ar ° ·

40
Curvilinear Coordinates
We can take as the Independent variable t, the abscissa s of a point on the
straight Hne, s calculated on this line to depart from a fixed origin, and
we have
This same result would have been obtained by finding the lines which minimize
the Integral /ds.
We owe to Lagrange a general method 1n theoretical mechanics allowing us
to determine the preceding result directly. Effectively, we form the
expression 2T which gives the square of the velocity of a point
2T = gjjCJjV)' .
The equations of motion of this point, assumed to be uninfluenced by any force,
are
d эт эт n
Э(и ) Эй
But the theory of Lagrange's equations 1n mechanics gives us something more
than just the differential equations for straight lines. The quantities
d = d ЭТ ЭТ
pi ЬТ 77177 ■ 7T
(Эй ) Эй
allow us 1n fact, to calculate the elementary work of the acceleration vector
for an arbitrary displacement 6u from the point: this work 1s
Yjdu = Р^би ;
the quantities P. are thus none other than the со variant oonponenta of the
aooeleration of any moving point.
The verification 1s easy. We first calculate the covariant components
directly by starting from the contravarlant components γ given by the
relationship 1n (19). By taking account of (II), we can readily obtain
A2 к Alk . h
1 1k ^7 k1h dt dt
On the other hand, calculating the quantity P. of the Lagrangian, yields

Curvilinear Coordinates
41
d - d In du\ 1 39kh duk duh
oU
„ d2uk + 1 Э91к + 39ih duk duh 1 3hb duk duh
B η d2uk + rth π duk duh
egik"T+ [кМ]-аг-аг ■
at
The two results are quite compatible.
38. The fundamental theorem of theoretical mechanics, a theorem resulting
from Lagrange's equations: the essential mechanical properties of a system are
virtually contained in the analytical expression of its effective force, seems
to embrace the fundamental theorem of metric geometry which 1s actually a
particular case of 1t.
From a more practical point of view, the algorithm of Lagrange 1s very
convenient for calculating the Christoffel symbols. In fact, for a given
index 1j the Г. .. are the coefficients of the quadratic form tfith respect
to the first derivatives (u ) which enter into the expression
о - d ЭТ ЭТ
For example, let us take the line element with respect to polar coordinates
r, θ, φ:
ds2 - dr2 + r2de2 + r2S1n2ea>2
Let us assign the numbers 1, 2, 3 to the variables r, θ, φ respectively,
We have
2T - r'2 + r2e'2 + r2sin2e φ'2
οΤ& - fff« A" ♦ an- e· - r2 sine cose φ'2
"3tэф' " эф и Αι^θφ" + 2r sin2erv + 2r2 sine cosee4'
With the exception of the zero Christoffel symbols, the result of this 1s

42 Curvilinear Coordinates
122 1| - -r 133 1| - -r S1n2e
|12 2| - r |33 2| - -r2 S1n2e Cose
|13 3| * r S1n2e [23 3| - -r2 Sine Cose
He can easily pass to the symbols of the second kind by dividing the equations
2 2 2
1n the first, second and third lines by 1, r and r S1nee respectively:
- -l I } - -r sin2e
-sine cose
221 |33
21, f2
121 r 133
3l .1 /3l .cose
13) r |23| ^Ш
39. The curvature of a space curve can be calculated without too much
difficulty. If the curvilinear coordinates of one of Its points are expressed
as functions of a parameter t which may be taken to be time, then 1t can be
shown that the Ыvector defined by the velocity and acceleration has for Its
v3
magnitude —, where ν denotes velocity and e the radius of curvature.
The velocity 1s given by
2 _ n du1 duJ
v " 9υ"αΤ"αΤ ·
the acceleration having been determined. Consequently, we can calculate the
magnitude of the Ыvector 1n question and deduce -. Let us just note that
once the equations of the curve are given, to determine the curvature at a
point Η only entails knowing the numerical values of the quantities g<<
к
and Г.. at that point, that 1s the g. . and their first order partial
derivatives.
Calculating the torsion, 1n contrast, would Involve the two first order
partial derivatives of the g...
On the other hand, given a surface whose equation 1s known, the normal
curvature of the different curves traced along this surface by one of Its
points Η Involves only the g.. and their first order partial derivatives,
and the same applies for the principal curvatures of the surface at that point,
the directions of the asymptotic tangents, the principal tangents, etc.

Curvilinear Coordinates
43
V. TENSOR ANALYSIS
40. The calculation made (see No. 34) to determine the components of the
absolute differential of a vector can be generalised to any tensor field. Let
us take as an example, a field of mixed tensors a. with two Indices. When
a point Η 1n space passes to a point Infinitesimally close to M, the
geometric variation of the tensor 1s an Infinitesimal tensor whose components
with respect to the natural frame attached to M, we shall denote by Daf.
To calculate these quantities, let us Introduce two arbitrary uniform vector
fields x. and χ and consider the sum
The elementary variation of this sum 1s evidently
on the other hand, by a direct calculation, 1t 1s equal to
da-jx1Y. + a^dX1Y. + *^X1dYj
By taking account of the uniformity of the two vector fields x. and jr, we
find
[da^ - ajiuj + *^]X1Yj
From this, the sought after relationship 1s easily deduced
Da^ - da-j - ajujf + ajfu)k1 (21)
It 1s easy to see what would happen 1f there were as many upper Indices as
lower Indices for a given number of them. If the preceding discussion 1s
applied to the fundamental tensor g.., we arrive at HIcd's theorem,
following which the absolute differential of the metric tensor is zero. This theorem
1s evident, since for two arbitrary uniform vector fields x. and jr, the sum
g,,X Xj 1s aonetant: 1t 1s the scalar product of the two vectors x. and jr.
Therefore we have
D91JX1XJ ■ 0
However, a proof of calculation 1s possible. He have
^U " dgu ■ 9кЛ " 91кш1

44
Curvilinear Coordivatee
and the right hand side 1s zero by the sane equations 1n (1) which had served
to determine the forms ω<. Calculating the absolute differential leads to an
12 η
Important result when applied to an η-vector a We have
n 12...n . 12...n л 12...η 1 л 12...1 J
Da = da + a ω. + ... + a ω1
<« ^J2...n j. ,12...η, 1.2. Λ η»
In - da + a (ακ + ω? + ... + ω )
If we assume 1n particular that the field of η-vectors 1s uniform, that
1s to say 1f we assume that the volume V of the n-vector 1s constant, we
obtain
da12-" d Jq dMl) 1 .
'9
we have then a significant relationship
У ■ω! - Нк*к <м>
which 1n fact could have been proved by a direct calculation of dg.
41. Absolute differentiation leads to a new operation in tensor analysis,
namely the derivation of teneora. The coefficients a. .. of the absolute
differential of a tensor a..
°*υ ' aijkduk ·
constitute 1n effect a new tensor. For evaluation, 1t suffices to show that
the quantities
b1j = a1JkX
define a tensor, where an arbitrary vector field χ was chosen.
To this extent, let us Imagine each trajectory of the vector field as a
trajectory being defined by the differential equations
du1 x du2 . dun
Ύ 7" ■" xn
described by a point 1n motion, 1n such a way that Its velocity 1s equal to the
corresponding vector of the field. We have then
The Index к is a derivation index and 1s sometimes distinguished from the
others by putting in a vertical bar and writing *jj|k Instead of a^.

Curvilinear Coordinates
45
b1j ""dt
a relationship evidently leading to the tensor character of b... He shall
note that the new Index к so Introduced 1s of а соvariant nature.
Let us apply this operation to a vector field X (contravarlant
components), or Xj (covarlant components). A tensor with two Indices 1s deduced,
having covarlant components X^. From the relationship
DX1
we deduce
xu
"dX1
эх1
k¥
■u1Xk
" r1JKk
The skew symmetric tensor
ХЛ
-xu
ЭХ.
3u7
■
•
эх1
ζ?
(23)
1s well known: 1t 1s the curl of the vector field. However, 1n calculating
directly the Ыlinear covarlant
αω(6) - oai(d)
of the invariant differential form
u(d) = Xjdu1 + X2du2 + ... + Xndun ,
we have
du(6) - Md) - —г (duJ6u1 - ouW)
3uJ
- i -4- -4 (du16uJ - duJ6u1) -
£ 3U 3UJ
The right hand side 1s the scalar product of the curl (considered as а
Ыvector) and the Ыvector determined by the two Infinitesimal vectors dH and
6M. Another Important tensor derived from a vector field 1s the divergence of
the field: 1t 1s the contracted tensor
X1
* 1 '

46
Curvilinear Coordinates
Now from the formula
1 1 It 4
OX = dX1 + XV
яэх; + хк ι
1 7T A 4ι
we deduce
du
By taking account of (22), we can restate the divergence under the guise of
d1Vx.Ml + ii^xk.iii5x!i . (24)
3u Jq bu /q 3u
This very straightforward expression can be obtained In another way. To
simplify Batters, let us restrict our attention to three dimensional space. In
ЭХ 3Y 3Z
rectangular coordinates x, y, z, the divergence ^ + ^- + %ь °? * vector
σΧ oy σΖ
field (X,Y>Z] 1s most ofte/i Introduced by calculating the flux of vectors
across a closed surface, on account of the relation
Xdydz + Ydzdx + Zdxdy = |£ + ψ + |j dxdydz - (25)
To transpose this relation Into any system of curvilinear coordinates,
let us consider an element of the surface of Integration as а Ыvector whose
plane 1s tangent to the surface and whose magnitude 1s equal to the area of
the element of the surface. This Ыvector must be oriented In such α way that
Its supplementary vector 1s 1n the exterior of the volume bounded by the
surface. The double Integral element which occurs on the left hand side of
equation (25) 1s thus the magnitude of the trlvector defined by the bivector under
consideration and the given vector (Χ,Υ,Ζ). As for the triple Integral
element on the right hand side, It 1s the product of the divergence and the volume
element of the space. The general expression
/ξ (X]du2du3 + X2duW + X3duW) - d1v χ ^ duWdu3
1s derived, and on applying Ostrogradsky's formula we can straight away see
that
divx.-lil^Oll . (24)
Jq Эй1
42. The study of fields of scalar tensors and their derived tensors leads
to some very Important concepts In geometry and Mathematical physics. A scalar
field 1s quite simply a function of points V(u ,...,un) defined Independently

Curvilinear Coordinates
47
of every frame of reference. The derived tensor
3u
1s the gradient of the function V; It thus defines a covarlant vector field.
The curl of this field 1s Identically zero. The magnitude of the gradient,
I.e.
JJ 9V ЭУ
du duJ
ie the firet order Beltrami differential parameter Δ, V- As for the
divergence of the gradient, this 1s the second order Beltrami differential
parameter:
V-F^'F A* 91k^ · (27)
c /ξ Эй /ξ Эй Эй*
In rectangular coordinates, we have
£ Ъх by Ъгс
In curvilinear coordinates for 3-d1mens1onal space, we have
Δ v B L· J^ (^ gU jv.,
/ξ эй эй
/9п9229зз 1Эи
/922933 ЭУ ) + Э /933911 ЭУ \ + Э /91192г\ ЭУ
I 911 Эй1/ Эй7! 922 Эй7/ Эй3! 933 /эй3
This formula 1s due to Lame.
We shall note that the divergence of a vector field, as well as both the
Beltrami differential parameters of the first two orders, only Involve the
first order partial derivatives of the components g.. of the fundamental
tensor.
In concluding, let us note that the second derived tensor V,, of a
scalar tensor 1s aymetria\ we have In effect
ν . »2v . rk JL
This property 1s however evident without calculation since 1n rectangular
(or Cartesian) coordinates, the components V., reduce to second order covarl-
2 1J
Э V
ant derivatives —j \ and the property that a tensor should be symmetric
3u'duJ

50
Curvilinear Coordinates
Ρ" P'" - PP'
which has αβΟ 0 χ for Its contravarlant components. Similarly, 1t can be
5 Г ^ ^
verified that the Infinitesimal vector p'p'" - PP" has for Its components
αβΟ D χ . He then deduce straight away that
°Λχ1 " °s°rxl B °
The calculation gives
1 aDrx1 к 1
DsV ■ "^r + 01s
32xi . Эхк г1 . „к ЭГкг . Эхк г1 . rj . yk rh .1
3ur3us 3us kr "17 3ur Pks Г" кЛ* "
The comparison with D D x leads straight away to the equations In (28).
We could represent this In the following manner.
The vector pp*1" can be calculated in two ways. Firstly It can be
regarded as the sum of the vector pp' and of the vector p'p'"; the first
term has for Its components αΟ^,χ1 and the second, coming from the vector
field pp", has for Its components
βυχχ* + αβΟ^χ1
We have therefore
pp"' - οΟγχΊ + 6Dsx1 + αβυ^χ1
On the other hand, on passing through the Intermediate point p" we obtain
pp,M ■ 605χΊ + aDrx1 + αβ050ΓχΊ
45. Instead of the two fields of elementary displacements
т*{6и - 0,...,6ur - a,...,6un - 0)
and
ММЧби1 - 0 6us - β 6un - 0)
we could consider two fields of arbitrary displacements that could be defined
In terms of the two interchangeable symbols of differentiation d and 6.

Curvilinear Coordinates
49
8%
By writing the second derivative —-f^-r 1n two different ways we have
3ur3us
whence the required conditions
<
3uS
эгк
dl1s . rh rk
" 3ur 1rhs"
?h rk
' 'Is'hr
1 (Uk.r.s - 1.2,....n) (28)
We shall replace the Г., 1n these equations by the values In (13) and thus
obtain the necessary conditions which must be satisfied by the functions g.,
1 η 1J
of u1 un.
44. These equations may be derived In an alternative way. Let us
associate to each point Η In the space, taken to be Euclidean, a point Ρ
defined by Its coordinates χ ,...,χη with respect to the natural frame
attached to the point M. We have, then, a field of points. The absolute
differential of Ρ 1s given (see No. 35) by
Dx1 «dx1 +du1 +xku^ . (17)
Let us put
D J . й1 + 1 + Λ
r 3(Jr r kr
(ej. - 0, If 1*r, z\ « 1 1f 1-r)
such that we have
Dx1 - Drx1dur
Now let α, Β be two Infinitesimal parameters (Independent of u ,...,
un). Given a point H(u] un), let us call H' ,H"tH,M respectively,
the points obtelned when firstly we Increase the coordinate u by a,
secondly Increase the coordinate us by В and thirdly. Increase
simultaneously ur by α and us by B. Let us denote by P,P,,P",P'", the points
of the field attached to the points H,H,,H,,,H,n, respectively.
The Infinitesimal vector ft' has the quantities αΟ χ for Its contra-
variant components with respect to the natural frane attached to H. These
quantities define a vector field; when we pass fron Η to H't the vector of
these field undergoes a variation

50
Curvilinear Coordinates
The
cal
Iculatlon
DsDrx1
32x1
gi
-
- +
ves
3Drx1
3uS
^ ,
P" P'" - PP'
which has αβΟ D x1 for tts contravarlant components. Similarly, Η can be
verified that the Infinitesimal vector p'p"' - pp" has for Its components
αβΟ D χ . We then deduce straight away that
DrD x1 - DsDrx1 ■ 0 .
+ Drx4s
r1 + xk ujcr + &* ri + pJ + xk rh 1
3ur3us 3uS kr 3us 3ur ks rs kr hs "
The comparison with D D x leads straight away to the equations In (28).
We could represent this In the following manner.
The vector pp1" can be calculated In two ways. Firstly It can be
regarded as the sun of the vector pp' and of the vector p'p'"; the first
term has for Its components aOrx and the second, coming fro· the vector
field pp", has for Its components
30χχ1 + <x6DrDsx1
We have therefore
pp'" = aDrx1 + 6Dsx1 + (хВиДх1
On the other hand, on passing through the Intermediate point p" we obtain
pp'" * BDx1 + <xDrx1 + aSDsDrx1
45. Instead of the two fields of elementary displacements
m'(6u =■ 0 6ur = a 6un » 0)
and
MM'W - 0 6us - 6,...,6un - 0)
we could consider two fields of arbitrary displacements that could be defined
In terms of the two interchangeable symbols of differentiation d and 6.

Curvilinear Coordinates
51
Letting the symbols D and Δ denote the corresponding absolute
differential operators, the required conditions are expressed by the relationship
ΟΔχ1 - ΔΟχ1
Now
1 - δχ1 + 6u1 + хкШ|[(б) ;
Δχ
ΟΔχ1 - d(Ax1) + Axku>|[(d)
= d6x1 + dfiu1 + dxk(uj(6) + xkdmj|(6)
+ 6xkiuJ(d) + 6u1u)j|(d) + xkuj(6)cuj(d) .
By comparing with ΔΟχ and noting that d6x - 6dx and d6u ■ 6du , we
obtain
xk{duj[(e) - 6u^(d) + |di[J(6)^(d) - toj|(d)uij(6)|}
+ <rih ■ rik»*k duh ■ ° ■
By virtue of the symmetry of the Γ* the required conditions are obtained
under the guise of
duij(6) - 6mJ(d) » m[J(d)ioJ(6) - u>[!(6)(D1h(d) (1,k-l n) . (29)
The left hand side of the equations in (29), are the oovariant bilinear forme
(see No. 41) of the expressions ω). On denoting them by dto?, the condensed
form 1s obtained
α ΐ ι h 11
<4 Kwhl
w
(30)
h 1
where the symbol |ω.ω.|» replaces the determinant
"k<d»
ω[|(δ)
-h(d)
ω^δ)
We shall return later to all of these notations (Chapter VIII, Section I).
Both sides of (30) will be Interpreted as the quadratic forms with exterior
multiplication (cf. 12 and II) constructed by the variables du ,du ,...,dun.

52 Curvilinear Coordinatee
1 1 *
By taking all the du zero save du ■ 1 and all the 6u zero, save 6u -1,
we arrive back at the equations In (2B) of section 43.
VII. EUCLIDEAN LINE ELEMENTS
46. Let us now approach the problem of knowing If the necessary condl-
2
tlons In (28) for a given ds In Euclidean space are sufficient.
For the noment we shall restrict our attention to part of the problem.
We shall consider a ds defined 1 η η variables u,...,un, whose
coefficients g J over a certain coordinate patch (D) are functions having
continuous first order partial derivatives, the discriminant being non-zero. To
2
cut matters short, we shall say that the metric defined by the ds 1s
regular throughout the domain (D). Lastly, we shall assume that (D) 1s aimply
In _
connected. This means that If the u ,u u are regarded as the Cartesian
coordinates of a point In ordinary n-d1mens1onal space, the coordinate patch
(D) 1s represented In this space by a domain whereby each closed curve traced
on It could contract to a point by a continuous deformation.
These hypotheses being flmly adhered to, we are going to show that If the
relations In (28) are verified, It 1s possible to represent (D) be a domain
Δ conveniently chosen from Euclidean space, whereby the square of the distance
2
between two points 1nf1n1tes1mally apart In Δ 1s equal to the given ds .
To this extent we are going to determine a point Ρ and vectors e. In
Euclidean space, of such a kind that we have Identically (see 33).
dP - du1^
(3D
d<4 ■ 4* ■
We shall add the following Initial conditions: for a system (u ) of
values of the variables (situated In the domain (0)) the point Ρ occupies
a given position Ρ and the vectors e. coincide with the giver vectors
(e.) whose mutual scalar products are equal to the values of the g., for
the values (u ) of the variables.
We are going to show that the system In (31) 1s compatible. Let
(u ).,(u ). (un). be any system whose values come from the domain (d).
If on Integration, the system admits a solution, we could obtain the point Ρ
and the vectors e. corresponding to this system, by taking In the domain
(D), a path taken from the (u1) to the (u ), and Integrating the
equations In (31) along this path. The u will be taken as continuously dlffer-
entlable functions of an Independent variable tt that reduce for example,
to (u ) for t»0 and to (u )- for t»l. The differential equations

Curvilinear Coordinates 53
dP du1 β
It = ST^\
(32)
dt In dt ^k
will be Integrated by taking for the Initial values of the unknowns the point
Ρ and the vectors (e.) . For fl, we will thus arrive at a point Ρ
and with vectors e, suitably determined. Besides, we can be sure that It Is
possible to Integrate throughout the interval Οέύέΐ, on account of the
linearity of the equations.*
47. Firstly, we shall prove that the vectors e_, obtained, satisfy for
each value of t, the relations
Effectively, we have, fron (32),
^V^ _ irk duhe +_k duh. ._ .
—3t^" (Γι^"3Γ^*^ + rjn-ar VV ■
The quantities e. · e. thus satisfy a system of differential equations which
1 J I.
admit as solutions the g.., following the same way that the Γ\^ (see No.
33) were defined. The two solutions e. · e. and g... whiah correspond to
the вате initial conditions for t-0t are thus Identical.
48. We are now going to show that If In the domain (0), a passage Is
made fron the system (u ) to the system (u h by another path, then we
arrive at the same point Ρ and with the same vectors e,, as by the first
path.
In fact, as the domain (D) Is simply connected, It Is possible, by a
continuous deformation, to pass fron the first path to the second. Let us
Imagine then a continuous family of paths depending on a parameter a and
containing the two given paths. Each path of the family could be defined by
the relations
u1 - f1(a,t)
and without loss of generality, we can assume that for t-0, we have
u1 - (u1»
E. Goursat, Coure d'Analyee mathematique\ 2 edition (Paris, Gauthler-
Vlllars, 1911), pp. 370-1.

54
Curvilinear Coordinabee
and for t г I,
u = (u )1
regardless of a.
By proceeding along each path of the family as we had just Indicated, we
shall have a point Ρ and vectors e. for each system of values of a and
t.
By hypothesis, we have
(32)
Now let
ЭР
9t '
*1
9t *
us put
ЭР
Эа '
Эе-1
Эа
Эи1
at
*1 *
■ Г1НТГ
эй1
"ЗГ
- гк
4h
Ь =
Эй"
Эа
0
*к"
ε
h'
0
Е1
Let us differentiate the first equations with respect to a, the second with
respect to t and subtract. We shall obtain
dt
dε.
4t'
Эй1 Э^ Эй1 Эе-1
dt Эа " Эа dt
к Эиь ^ 9uh ^ arfh
Ih dt Эа Эа dt .1
dul
Эа
Эй
dt
Эй1
dt
UL e
Эа ^к
(34)
de, de.
Let us now replace the ~- and -^- by their values derived from (32) and
(33). The relations In (28), assumed verified, Indicate precisely that the
right hand side of the equations tn (34) are zero. If the ε and e, are
taken to be zero. The equations In (34) thus reduce to the following:
dt ' dt E1
9E1 . rk 9uh F
~W r1h ~W Ek ■
For t-0, the point P, the vectors e, and the functions и (t,a) do
not depend on a; consequently the vectors ε and ε, vanish when t* 0.
As they satisfy the differential equations In (35) which admit the solution

Curvilinear Coordinates
55
ε ■ ε, ■ 0, we have Identically
с ■ с, - е- ■ ..." с ■ О
ι с η
Эй,
The -gj- are zero by hypothesis, for t»l; the relations In (33) show that
for t-1, we equally have
3a u · 3a
The point Ρ and the vectors e, do not, therefore, depend on the parameter
a for t= 1; this Is what we wished to prove.
49. The point Ρ and the vectors e. are functions determined by
u ,...,un. Given two systems of values, u and u +du infini tesimally
close, It can be assumed that the two corresponding systems (P.e,) have
\ '
been determined by means of a path leaving the (u ) and passing through
111
the u and u +du . This proves that the equations In (31) are Identically
verified. The vectors e, constitute the natural frame of the Euclidean
1 \
space with respect to the coordinates u and, as In 47,
*
2 2
the ds of the space Is Identical to the given ds .
The rectangular coordinates χ of the point Ρ which corresponds to
a system of values u are the determined functions
x1 ■ fV un) (36)
It Is easy to see that the functional determinant of the F Is never zero.
The equation
(dx1) - QjjduW
shows In effect, that we have
к к
Эх Эх* в п
τ?τ? '" '
Calculating the square of the functional determinant of the χ with respect
to u straight away gives the value g. The discriminant g having been
assumed nonzero throughout the domain (D), the result Is the stated property
of the functional determinant.
*
See In note V, another way of showing the compatibility of the system In
(31) which Is known as a completely integrable system.

56 Curvilinear Coordinates
He cannot conclude from this that each system of values of the χ given
by the equations In (36) corresponds to a single system of values of the u ;
we can only do this If we restrict our attention to a sufficiently small
neighbourhood about the system of values (u ) In the doeain (D).
The result of this is that to the domain (D), there corresponds a
domain (Δ) in Euclidean apace admitting the given ds , but this domain
(Δ) aan be covered partially or totally. If (D) is restricted to a
sufficiently email neighbourhood of the (u ) , the aorreeponding region (Δ) is
not oovered, and there is a one-to-one correspondence between the considered
system of values of the u and points of (Δ). The region (Δ) is sirrply
connected, as is the corresponding domain (D).

Chapter III
LOCALLY EUCLIDEAN RIEHANNIAN SPACE*
I. THE CONCEPT OF A MANIFOLD
50. The general concept of a manifold is quite difficult to define
precisely. A surface gives some Idea of a two-dimensional manifold. If we take,
for example, a sphere, or a torus, we can divide this surface Into a finite
number of regions such that there exists a one-to-one representation of each
of these regions on a simply connected region of the Euclidean plane.
More precisely, given any point Ρ on the manifold. It Is possible to
find In a neighbourhood of Ρ , a co-ordinate system uyv such that If u ,
у are the co-ordinates of Ρ , there exists r > 0, having the following
property. Every system of numbers u,v satisfying the Inequality
(u - uQ)2+ (v - vQ)2 <p2 (1)
constitutes the co-ordinates of a point and of a single neighbourhood of Ρ
on the manifold. Conversely, in a sufficiently small neighbourhood of Ρ ,
every point Ρ has co-ordinates u,t> satisfying the inequality In (1).
The sphere and torus are 2-dimensional manifolds without boundaries. A
eyeUnder of revolution, a hyperbolic paraboloid are 2-dimensional open
manifolds (with boundaries at Infinity). A sheet of a cone of revolution, the
vertex excluded, constitutes a manifold which has a boundary at Infinity and
a boundary at a finite distance (the vertex).
The volume contained Inside a sphere constitutes an open, 3-dimensional
manifold, the boundary being the surface of the sphere. When this volume
Includes the surface, It constitutes a 3-dimensional manifold with a boundary,
but the boundary forms part of the manifold which is said to be closed. In
the preceding examples, each manifold Is defined by a set of points situated
In a pre-existent space. But these manifolds may be considered in abetracto.
*
Concerning the material dealt with in this chapter, we might consult:
H. Killing, EinfWirung in die Grundalagen der Gaofrwtrie, t. 1, Paderbron 1893;
F. Klein, Conferences виг lee Natk&natiquee faitee h I'Exposition de Chicago
(Conf. XI); J. Hadamard, Sur la forme de I'eepaae (Proc. verb des seances de
la Soc. des Sc. phys. et nat. de Bordeaux, 1897-1898, pp. 83-85; H. Ueyl, Die
Idee der Riemcmneehen Ftaohe(Leipzig und Berlin, 1923 and also Math. Ann.,
t. 77, 1916, p. 349); H. Hopf, lien Clifford-Kleinechen Rawproblem {Math. Ann.,
t. 95, 1926, pp. 313-339). See also F. Enrlques, Principee de la geometric
(Encycl. Sc. math., t. Ill, 1, pp. 131-136).
On the general concept of a manifold, we might consult: F. Hausdorff,
Grundzuge der Mengenlehre (Leipzig, 1914); P. Alexandroff und H. Hopf, Topolo-
gie, I (Berlin, 1935); N. Bourbakl, Elements de Mathimatique, Llvre 111,
Topologie generate (Paris, Hermann, 1940).
57

58
Looally Euclidean Riemannian Space
In the general case, an π-dlmenslonal manifold Is characterised by the
possibility of representing the neighbourhood of each point Ρ by means of a
system of η co-ordinates ul that may take all possible values In the
neighbourhood of the system of values (и*) which represent Ρ .
51. The co-ordinates susceptible to analytical representation of a region
of a manifold can be chosen In an Infinity of ways. On passing from one
system of co-ordinates to another, It Is understood that the new co-ordinates are
continuous functions of the original and vice versa. The "analysis situs" here
Is concerned with the study of those properties of manifolds that are
Invariant under such changes in co-ordinates.
In differential geometry, there Is an extra condition that the new
co-ordinates considered as functions of the original are not only continuous,
but also admit continuous partial derivatives up to a certain order. The field
of properties that are invariant under such changes In co-ordinates Is already»
oddly enough, more extensive than the "analysis situs."
Let us define, forexample, a line giving the co-ordinates of Its points
as functions of a parameter t. To say that the functions are dlfferentlable
with respect to t. Is to state a property of the line which is preserved
under each change of admissable co-ordinates. We thus arrive at the Idea of
a line element. Analytically, a line element Is defined by η co-ordinates
u ,...,un and the mutual products of the differentials du ,...,ώ*π.
Geometrically, It is defined by the set of lines tangent to each other at a given
point.
By considering the set of line elements stemming from the same point and
1 я
which satisfy the same system of (n-2) linear equations In du ,.,.,ώ ,
the Idea of a plane element Is obtained. Evidently, It Is a property of this
set of line elements that is preserved under a change of admlssable
coordinates. Similarly, a plane element Is defined In 3,4 etc., dimensions.
If stemming from a given point there are 4 line elements tangent to the
same plane element, then the cross-ratio of these line elements Is a number
which Is the same under any change of co-ordinates. One Is able to generalise
these matters.
On the whole It could be said that the study of the properties of this
nature Is the geometry of the manifold from the point of view of the group of
continuous and differentiable point transformations, whereas the "analysis
situs" Is the geometry of the manifold from the point of view of the group of
simply continuous point transformations.
If we assume that the new co-ordinates admit partial derivatives of the
two first orders with respect to the originals and conversely, then the field
of geometric notions extends accordingly, we could say that lines have contact
of the second order with each other.

Locally Euclidean Riemannian Space
59
52. A Riemannian manifold Is a manifold to which a metric Is attached.
This means that In each region of the manifold, analytically represented In
terms of a system of co-ordinates ul, a quadratic differential form Is given»
de = g.du duJ
We shall assume that the д.. are continuous functions admitting continuous
partial derivatives of the two first orders. Consequently we shall only admit
those changes In co-ordinates such that the new co-ordinates admit continuous
partial derivatives of the two first orders with respect to the original со-.
ordinates, and conversely.
We shall say that the metric 1s regular In a given region of the manifold,
2
If at all points of this region, the de Is a positive definite form of the
Naturally enough, we shall assume that If the manifold consists of several
regions admitting distinct analytical representations, a connection between the
metrics for the neighbouring regions Is possible. It could be assumed, for
example, that the analytical representation of each region could be extended
2
by a small amount Into one of the neighbouring regions, and that the two de
thus obtained In the overlapping regions, are each reducible In terms of the
other, by the change of co-ordinates which takes one analytical representation
to the other.
II. LOCALLY EUCLIDEAN RIEMANNIAN SPACE
53. A Riemannian manifold Is said to be locally Euclidean If In each
region of this manifold, defined analytically by a system of co-ordinates
и1, the de satisfy the conditions In (28) (no. 43) of the Euclidean space
line element.
On account of what had been proved at the end of the last chapter, this
means that the manifold, In a sufficiently small neighbourhood of any one of
Its points Η can be represented in a small domain in Euclidean space with
2
the same de . We shall say that this representation constitutes a
development of that region in question of the manifold. In Euclidean space.
Conversely, the domain obtained from the Euclidean space Is developable on the
corresponding small region of the manifold.
If the metric of the Riemannian manifold Is everywhere regular, we may
see that it Is possible to develop all of the manifold piece by piece in the
Euclidean space. But It Is not for certain a priori:
1. That each point of the Euclidean space could be obtained in the
development.
2. That a point of the Euclidean space obtained In the development of
the manifold could not be so obtained more than once.

60
Loaalty Euclidean Riemamian Space
54. Before going any further, let us clarify what this means by some
simple examples taken from the two-dimensional case.
A cylinder of revolution embedded In ordinary space, has for Its line
element
da2 - du2 + do2
where и denotes the curvilinear abscissa (0 < u < l) taken along a straight
line section, and u, the ordinate. This line element Is Euclidean; but the
manifold constructed by the cylinder Is not simply connected, and the
development In the Euclidean plane gives a collection of Infinite strips of width I.
Each point of the cylinder corresponds to an Infinite number of points on the
plane that are deduced, one from the other, by a translation with a fixed
direction and whose length Is an arbitrary multiple of I. We shall see that
here the plane ie once covered entirely* An Image of the manifold Is obtained
by taking In the plane, an Infinite strip In both directions, having a width
I and In identifying opposite points on the limiting parallels when the
straight line joining them Is perpendicular to these parallels.
Another example Is provided by a torus; the position of a point on a
torus Is completely determined by two angles θ and φ, 0ί θ, φ < 2*. By
assigning a line element to the torus
da2 - ad&2 + 2Ь<Йф + саф2 ,
with constant coefficients, a Euclidean metric Is defined on this manifold.
On the Euclidean plane the variables θ and φ will be Cartesian co-ordinates.
The torus develops In the Euclidean plane as a parallelogram 0 £ θ < 2w,
0 s Φ < 2π, as far as It Is allowed for defining the development, by only
tracing on the torus, the lines not crossing the line Θ-0 nor the line
Φ ■ 0. If these restrictions are taken away, the torus develops In all of the
plane which Is covered once and only once; but the domain representative of
the manifold is a parallelogram whose opposite sides are not regarded as
distinct.
A last example Is that of an unlimited sheet of a cone of revolution
whose da results from the metric of the space (ordinary) In which It Is
2
embedded. This da Is also Euclidean, as we know. Here It so happens that
the vertex of the оспе ia α aingular point of the metric, since a half-line
(straight) stemming from the vertex and passing successively through all
directions (on the cone) deacribee on angle leea than 2т. If we wish to
*
Following a remark by U. Killing, we have a concrete Interpretation of this
manifold, for b ■ 0, by taking In four dimensional Euclidean space, the
surface defined by the equations x, - /a cos Θ, x- * & s**n θ» хз * & coS ♦>
χ. = Jc sin φ. Clifford has given another Interpretation In 3-dlmenslonal
elliptic space.

Locally Euclidean Riemannian Spaae
61
avoid considering the singular points of the metric, we must exclude the
vertex of the manifold under consideration. It will therefore be open on the side
of the vertex (ami on the side of infinity) so becoming from the topological
viewpoint, Identical to a cylinder of revolution.
The development of this manifold on the Euclidean plane will now give all
of the plane {uith the exception of a point), but thie plane ie oovered an
infinite number of timee (at least If the sine of the half-angle at the peak
Is an Irrational number). As It can be seen the result Is quite different
from that obtained In the two preceding cases.
2
Finally, any developable surface also has an Euclidean de , but where
1t folds, there are singular points of the metric. By taking only one of the
sheets of the surface, we obtain a development which covers one region only
of the Euclidean plane* and could cover this region several times and likewise
an Infinite number of times.
55. On account of this, there seems to be a correlation between the case
of an entire covering of the Euclidean space and the case of covering just
once. The last two examples quoted have this in common. I.e. the manifold
in queetion ie open to a finite dietanae, a situation which does not arise In
the first two examples (the cylinder and torus).
He are going to exclude from our discussion those Riemannian spaces which
present any similarity to that of the cone. It is necessary In this case to
consider them intrinsically, and not with respect to any pre-existent ambient
space that might contain the».
Let us firstly define the distance [AB] between two points A and Β 1η
a Riemannian space with an everywhere regular metric, as the lower bound of
the length of the curve arcs (rectlflable) which join the point A to the
point B. We can see quite easily that for any three given points A, B and
C, we have thp Inequality
[AC] s [AB] + [ВС]
He shall call a ephere centred at A and of radius R, as the set of points
Μ satisfying the Inequality
[AM] s R
An Infinite set of points of the space will be said to be hounded If the
distance of a fixed point A to the points of the set Is bounded; this property
Is evidently Independent of the fixed point A chosen.
He shall say that a point Ρ of Riemannian space Is a limit point of an
Infinite set (E) of points of this space If, In every sphere centred at Ρ
and of arbitrarily small radius r, there exists at least one point of the
set distinct from P, there then exists an Infinite number of them.

62
Locally Euclidian Riemannian Space
56. Having established these definitions, we shall only consider
Riemannian spaces with an everywhere regular metric enjoying the property that
every infinite, bounded point eet of this epaae admits at leaet one limit
point. This property Is usually expressed by saying that the oetrlc
Is complete. It Is evident that an unlimited sheet of a cone {vertex
excluded), considered as a 2-dimensional Riemannian space endowed with a
metric that Induces a Euclidean space [ordinary) In which the cone Is
embedded, does not share the preceding property. We shall say that a
Riemannian space with an everywhere regular metric having this property Is
normal. The cylinder, the torus (with the metric defined as above), and
Euclidean space Itself are evidently normal.
There are two main classes of normal spaces.
A normal Riemannian apace will be eaid to be closed, or compact, if every
infinite eet of points admits at one limit point. In such a space the
distance [AH] of a fixed point A to a variable point Η Is bounded;
otherwise there would exist In fact an Infinite number of points Η,,Μ^,.,.,Η ,...
such that the distance [AH ] may be continued Indefinitely. But this Is
ч
Impossible, for such a collection would acinit at least one Unit point Ρ
and, In the Interior of a sphere centred at Ρ and of radius r, there would
exist an Infinite number of points and therefore points И at an arbitrarily
large distance from A, whereas we have
[AHn] S [AP] + r .
We might add that the distance [NN] between two variable points Is equally
bounded, by virtue of the Inequality
[MN] s [AH] + [AN]
Conversely, It Is clear that If the distance [HN] between two variable points
of a normal Riemannian space Is bounded, the space Is closed, for every
Infinite collection of points of this space Is bounded and consequently admits a
limit point.
It could be said that a normal Riemannian space which Is not closed ia
open to infinity. This Is thus the case with a cylinder of revolution and
Euclidean space Itself. This expression Is self explanatory: It expresses
the existence of Infinite collections of points moving Indefinitely away from
a given point A without having a limit point.

Locally Euclidean Riemannian Space
63
III. LOCALLY EUCLIDEAN NORMAL, RIEMANNIAN SPACE
57. We now Intend to prove the following fundamental theorem. If a He-
marovian βρασβ with a Euclidean metric га normal^ it в development in Euclidean
epace covers all of this βρασβ once and once only.
We have seen (see No. 49) that given a point Μ In any Riemannian space,
there exists a positive number r such that, If corresponding to Μ we take
a point Ρ In Euclidean space, there exists a one-to-one correspondence
between the points of the sphere (S) of centre Ρ and radius r, and the
points of the Riemannian space which are In a particular neighbourhood of Μ .
These points evidently generate a simply connected sphere (Σ) of centre Μ .
The number r attached to the point Μ Is, naturally enough, not
determined uniquely; It could be replaced, for example, by any smaller positive
number. But the following statement Is fundamental:
If the point Μ' ie a distance from Μ leee than a positive number
ε < r, we could attach to M" the number r' ■ r - ε. In fact, the point
M* being Inside (Σ)> there corresponds to It In Euclidean space a point
P' Inside (S), and the sphere (S1) with centre P1 and radius r-ε Is
completely Inside (S). This assures the existence of a one-to-one
correspondence between the points of (S") and those of a certain neighbourhood of
58. We are going to deduce from the preceding remark, that If the point
Μ stays Inside or on the boundary of a bounded domain, for example a sphere
of Riemannian space, the quantity r stays greater than a fixed number p>0.
If In fact that were not the case. It would be possible to find an Infinite
collection of numbers
ε1,ε2,...,εη,...
tending to zero, as well as an Infinite collection of points of the domain
MTM2 Mn·'"
such that It Is Impossible to attach to the point Μ , a number r greater
than ε . Now the collection In question would at least adnlt a limit point
Μ . to which we can attach a number r . If ε denotes a number taken to be
о о
as small as we wish, then there would be an Infinite number of points In the
collection Μ which would be a distance from Μ less than ε. To each of
η о
these points, the number r - ε could be attached, which manifestly ends by
exceeding ε for η sufficiently large.
Having proved the proposition, let us take, In Riemannian space, a point
Μ as an origin, to which we shall correspond In Euclidean space a point Ρ
as well as a Cartesian frame (R ) defined to within an orientation by the

64
Locally Euclidean Riemannian Space
numerical values of the coefficients д.. of the fundamental font In Μ . We
IrJ 0
can note moreover that (R ) is defined by a rotation and to vithin a
symmetry. It Is from the point Ρ , with Its frame (R ) that we are going to
develop the Riemannian space In Euclidean space.
We shall prove successively two theorems which together will constitute
the above mentioned fundamental theorem.
Theorem. Euclidean apace ia covered completely in the development of the
Riemannian Space.
59. Effectively» let Ρ be any point In Euclidean space. Let us join
Ρ to Ρ by any line (C) of finite length I. We shall consider In the
Riemannian space the sphere (Σ) with centre Μ and radius R, where R
Is a given number, R>Z. (This sphere could be Identical to the Riemannian
space Itself, If this Is closed and R Is sufficiently large.) Corresponding
to (Σ) Is a positive number ρ less than all the numbers r attached to
different points of (Σ1 .
Let us Inaglne, In Euclidean space, a collection of spheres radius p,
the first of which we take to have a centre Ρ ,' the last having Ρ for Its
centre; the Intermediate spheres will have their centres on the line (C) and
will overlap such that each point of (C) Is an interior point to one of the
following spheres. The line (C) could also be divided Into a finite number
of arc segments,
PoPTPlP2 Pn-lP ·
such that each of them is situated Inside one and the same sphere of the
collection.
With this established, the arc Ρ P|, situated Inside the first sphere
of radius p, could be developed In the Riemannian space following an arc
MM, completely contained in (Σ); the sane Is true of P,P2 and the
successive arcs that will be developed In the Riemannian space without ever going
outside of (Σ)· The line (C) will thus develop completely a following
certain line (v), which conversely would give (C) In the development of
the Riemannian space in the Euclidean space. The theorem Is thus proved.
60. At the extremity Μ of the line (v), the д.. have given numerl-
cal values. If It Is the same system of co-ordinates which sets in the
Riemannian space the entire length of (v), there will be attached to the
different points of (C) Cartesian frames varying continuously, such that
each point of (C) will be assigned a frame R.
If there are successively two distinct systems of co-ordinates (u )
and (уг) in those regions of the Riemannian space traversed by the line (v),
we will have on the first portion of the line (C), a Cartesian frame (RJ

Loaally Euclidean Riemannian Space
65
then on the second, a Cartesian frame (R ). At the point of separation, there
will be a discontinuity In the variation of the frame. It could happen,
however, that the line (v) from Μ of the region with co-ordinates цг, then
crosses the region with co-ordinates υ" and then returns to the region with
co-ordinates u\ Thus It will be possible to have on (C) an arc with frames
(Ru) a second with frames (R^) and a third with frames (R ).
When the same system of co-ordinates u1 are Imposed on all of the curve
(v), the frame (R) attached to Ρ has evidently the same orientation as the
frame (R ) attached to Ρ . But If the curve (v), leaving a region with
co-ordinates ul, returns to this region after having crossed a region with
co-ordinates ul, the frame (R ) attached to Ρ could not have the вате
orientation as the frame (R L attached to Рл. In fact to the first point
и Ό о r
of separation of the two regions, the two frames (R ) and (R ) will have
the same orientation or not, following the sign of the functional determinant
0(u\u2 un)
0(v\vZ vn)
To the second point of separation, the sign of this determinant will be
Involved again, but there Is no reason a priori, to assume for the two points
In question, that these signs are the same. Finally, let us clarify matters
by saying that a path stemming from Ρ In Euclidean space can only proceed
to a single path stemming from Μ In Rlemannlan space; this results from the
same proof which Is about to be made.
Theorem. Euclidean epaae ie covered once by the development of Riemannian
apace.
61. It suffices to show that two distinct lines (C) and (C) leaving
the point Ρ and ending at the same point Ρ develop on the Rlemannlan
space following two lines leaving Μ and ending at the same point H. The
proof reete on the property of the Euclidean apace being eixtply connected.
We can Imagine a collection of paths
(c),(c1) (cn-1),(c)
all stemming from Ρ and all of them ending at P. Let us denote by I and
V the lengths of the two paths (C) and (C) and we shall take on each
of them a parameter t varying continuously between 0 and 1 when the path
In question is described. For example, we could take
t = j on the first path
and
t ■ 7, on the second.

66
Looally Eualidean Riemamian Spaa*
where в Is the curvilinear abscissa taken fron Ρ . Taking rectangular
co-ordinates, we have
*i -/£(0 (i)
and
*, * iAt) (2)
the respective equations of the two curves. Let us define a curve (C ) by
the equations
x. 'ofAt) + (l-a)e.(t) (0<a<l) . (3)
We have on (C ),
a
ds2 - [<A2 + (l-a)2f2 + 2a(l-a)/jUW£(0]*Z
Now the Inequality
|^(t)*:(OI s ^(^ ^*^ " "*
gives
<ie2 S [at + (1-α)£']Ζ<**2 .
Letting L = яах{£,£'}. It can be that that the ourvilinear abedaea of arty
point of (С } гв at moat equal to Lfc, and consequently, all of the curvee
of the family, are of length at most L.
Letting R>L. let us consider In the Rlemannlan space, the sphere \Σ)
with centre N and radius R, and let ρ be a number less than each number
r attached to different points of (Σ)· Ρ > 0.
Finally, let us evaluate the distance б between two points on the two
curves (C ) and (C) corresponding to the same value of t. Me have
б2 - (a - α')2 Σί/£(0 - *£(*)]Z ·
2
Let D be the me χ 1 пили value of the summation on the right hand side, when t
varies between 0 and 1. We obtain
iS(a-a')D .
With this established, let us divide the Interval (0,1), In which α varies.
Into Intervals each less than % ij- and consider the curves (C.),...,(C ).
(C) corresponding to values of the subdivision. In terms of t9 let us
divide (0,1) 1n exactly the same way and let 0,t, ,Ц»... ,t _1»1, be the

Looally Sualiaean Riemannian Space
67
points of the subdivision. To these values of i, there corresponds on each
curve (C^), the points Ρ ,P and p-1 Intermediate points ?t P{p" .
The epheree of radius ρ and oentree
Ρ Ρ1 Ρ2 Ρ?-1
Ио· VYi ΐ
have the property that every point whether it ie on (C.) or (C.+i) ΐβ
contained within one of theee spheres.
This Is evident for f^. Let Q be any point on Ct-+1 (Fig. 1) and let pj+1
be that point from the ρ£+, points, that Is the nearest to Q. We have
« <i«Η fL · ι* ·
and on the other hand
Λ <!§»■!' ·
consequently
QpJ < yp+ jp ■ ρ .
In other words, the point Q Is an interior point of the sphere centred at
A
г
To the p + 1 spheres of radius ρ having their centres on C, there
corresponds In the Riemannian space (p + 1) overlapping spheres whose centres
are not outside (Σ)· Tne curve (C) produces In the Riemannian space a
curve (v) leaving Μ and ending at a certain point M, the centre of the
last sphere. The curve C, will also develop following a curve (v1) leaving

68
Locally Euclidean Riemannian Spaae
Μ to end at M. By considering the (p+1) spheres having their centres on
C,, It can be shown In a similar way that the curve C? will develop
following a curve v- from Μ to M. We can then see that bit by bit, the curve
(C), also In the Riemannian space, will give a line (v1) from Η to M,
and yet completely Inside \Υλ· The result of this is that in the inverse
development of the Riemannian space, it ie always the вате point Μ that comes
to occupy the poeition P. This Is what we wished to prove. The fundamental
theorem Is thus firmly established.
62. Let us assume In particular that the Riemannian space Is simply
connected. The development of the Euclidean space on the Riemannian space covers
It once and only once. Consequently, the two spaces are identical. In the
sense that there Is a one-to-one point correspondence between points of the two
spaces ι with distances preserved.
Every normal, simply connected Riemannian spaae with a Euclidean metric
is idential to a Euclidean space.
In particular, such a space extends to Infinity. It results that for
η «2, the surface of a sphere, which Is Simply connected, could not be
analytically defined in terms of a single, everywhere regular co-ordinate system.
Otherwise, If и and ν denote the co-ordinates, the line elements, de *=
2 2
du *dv would define on the sphere an everywhere regular Euclidean metric.
IV. THE H0L0N0MY GROUP OF A NORMAL, LOCALLY EUCLIDEAN RIEMANNIAN SPACE
63. Now let us take a normal, locally Euclidean Riemannian space, but one
that Is not simply connected. The point Μ gives, In the development of the
Euclidean space, several (or an Infinite number of) points
Ρ0,ΡΓΡ2""
each endowed with a Cartesian frame
(R0).(R1).(R2)·---· respectively
corresponding to the same numerical values of the д.. at the pofnt Μ . All
of these framee are equal or eymrntria.
Clearly, If the development had commenced by corresponding to the point
Μ , the point Ρ* with the frame R,, the development of the Riemannian
space would not have undergone any essential change. If a certain path υ
going from Μ to Μ had been taken In the first development to a point Ρ
with a frame (R), the new development would lead to a point P1 and a frame
(R1) aitunted with respect to (R.) in the вата way as Ρ and (R) were
situated with respect to (Rn)·

Locally Euclidean Fiemannian Spaoe
69
This results In the displacement $1 (whether or not accompanied by a
symmetry), which In Euclidean space brings (R ) to coincide with (R),
varying between the frames
(RoMR,).^).... .
In effect, the frame (R£) Is deduced from (RQ) by the development of
a certain closed contour (γ.) leaving Μ and returning there. The develop-
ι» Ο
went of the same cycle leaving (R,) will give a certain frame (R.) situated
relative to (R,) as (R.) Is to (R ). This frame (R.) Is, In fact, the
110 J
one that would have been obtained In firstly developing the cycle (γ,) and
then (Yi).
64. The above considerations also show us that the displacements
S1»S2,S3·*··
forma group. In fact, by keeping the notation of the preceding section. It may
be seen that If successive operations are carried out on Sj and S{> the
frame (R ) firstly coincides with R, and then with R.; the resulting
displacement Is thus $.:
J
The group G» thus obtained Is called the holonomy group of the Rlemennlan
space. To each frame R.t there corresponds a single operation determined by
this group. I.e. one which takes (R.) to coincide with R.. To the frame
О Ъ
(R ) Itself there corresponds an identity operation.
The operations of the holonomy group may be applied to any point Q In
Euclidean space. If, for example, the operation S, Is responsible for taking
the point to Qj, then the point Q1 Is situated relative to (Rj) as Q
was relative to R . Let γ be a path traced In the Rlemannian space that
develops along a given path С going from Ρ to Q, and let N be the
extremity of this path. It will be seen that the point Q1 will be obtained
In developing the path formed by the cycle (γ.) and the path (γ). This
path so described leads from Μ to end at N; consequently the points Q,
and Q correspond to the same point in Riemannian space. We shall say that
they are homologuea.
The result of this is that every operation of the holonomy group sends
any point from Euclidean space to a homologous point. At the same time, we
note that there Is always an operation of this group sending any point of the
space to every homologous point.

70
Locally Euclidean Riemannian Space
65. We are now going to prove two fundamental properties of the holonomy
group.
Let Ρ be any point In Euclidean space that corresponds to a point И
determined In Rlemannlan space. To this point, a positive number r(r>0)
may be attached, such that every point P1 distinct from Ρ and situated at
a distance from Ρ less than r, arises from there being a point M1
distinct from M. Consequently, the homologuee of the point Ρ are all at a
distance greater than r from P.
This property expresses the discontinuity of the holonomy group. A group
Is said to be discontinuous If at ewery point Ρ there corresponds a number
r(r>0) such that all transformations of P are at a distance greater than
r from P.
It may be possible that for a discontinuous group, there exist certain
exceptional points that are themselves their own proper homologues for a
certain number of transformations of the group. Here, this cannot come about.
In other words, It may be said that every operation of the ЬоХопощ group,
other than the identity operation, changes every point of the epaoe.
We thus arrive at the following theorem.
Theorem. The holonomy group of a locally Euclidean Riemannian epaoe, ie die'
continuous ana each of its operations, other than the identity, changee a
point of the space.
V. THE FUNDAMENTAL POLYHEDRON
66. When developing a cylinder of revolution on a plane, ewry point of
the cylinder Is represented once and only once on a certain strip of the plane,
bounded by two straight parallel lines, whose length Is equal to the
circumference of the cross section of the cylinder. We are going to show that, In
the general case. It Is always possible to construct a polyhedron containing
one and only one representative of each point of the Rlemannlan space. We
shall consider the case of n-3 to fix Ideas.
It Is possible to construct such a polyhedron by ruling. Let us consider
the set of points of Euclidean space which are nearer to the point Ρ than
any one of the points Ρ,,Ρρ.-·-» homologous to Ρ . This set defines a
domain p (a fundamental domain) whose boundary will be formed by points
equidistant from Ρ and from one of the homologous points.
The domain p Is convex* for If Q and Q' form part of It, the two
points are situated on the same side as Ρ with respect to the perpendicular
plane In the centre of PP. (whatever value t takes). It Is thus the same
for ewery point R of the segment QQ*.
This being the case, let Q be any point In Euclidean space. By virtue
of the discontinuity of the holonomy group G, there exists one or several of

Locally Euclidean Riemannian Spaoe
71
the points Ρ ,P1.P2...·. nearer to Q than all the others. Firstly, let us
assume there 1s just a single P.. The operations S~. of the group G,
taking P. to Ρ , takes Q to a certain point R which 1s evidently nearer
to Ρ than all the other homologous points and Is subsequently contained
within the domain p. On the other hand. It Is not the case In this domain
that another point R1 should be homologous to Q, otherwise the operation
In G which takes R1 to Q would take Ρ to a point P. nearer to Q
0 3
than all of the other homologous points and consequently. It Identifies with
P.; likewise R1 would then Identify with R. If there are several such
points P.,P. at an equal distance from Q and nearer to Q than all of the
ь 3 -1-1
other homologous points, the operations S. ,S. would take Q to two points
*■ з
R and R1 respectively, the belonging to the boundary of p and homologous
to each other.
67. The fundamental domain p is a polyhedral volume bounded by a finite
number of plane faces, as we shall prove.
Let us assume first of all that the domain p Is of finite extent. Let
R be the maximum distance from Ρ to the boundary of p. Every point of the
sphere centred at Ρ with radius R, a sphere containing p In Its Interior,
Is clearly nearer to Ρ than the points P., in the exterior of the sphere
О 1r
centred at Ρ and radius 2R. It thus suffices to construct p by only
considering the points P. which are within or on the surface of this latter
sphere. These points are finite in number. Consequently p would be bounded
by a finite number of plane faces situated In the planes equidistant from Ρ
and from a certain number of the points P..
Now let us assume that p extends to Infinity. This means that there Is
at least a half-line stemming from Ρ and contained entirely within p
(because of the convexity of p). If Pz Is such a half-line, ewery point
P. homologous to Ρ Is evidently In the plane (n) perpendicular to Ρ я
taken through Ρ , or on the side of this plane opposite Pz.
First of all, let us assume that there Is a single half-line contained
entirely In the Interior of p. The distance Ρ to the points of p which
are In the plane (n), or on the side of (Π) opposite Pa, Is bounded above
by R. It then suffices to construct p by considering points P. situated
1n the Interior or on the surface of a sphere centred at Ρ and of radius
2R, and 1n each case, the same conclusion 1s reached.
In the second case, let us assume that there are two half-lines opposite
Pqe and Ρ a\ both contained entirely within the Interior of >. All the
points P. are necessarily In the perpendicular plane (n) common to Ρ ζ
ь о
and Pz1 taken through Ρ . It will suffice to consider those which are
Inside or on the surface of a sphere centred at Ρ and of radius 2R, where
R denotes the upper bound of the distance from Ρ to the points of p
situated In (Π). In this case p Is an Indefinite prism In both senses.

72
Locally Euclidean Riemxmian Spaas
Finally, let us assume that neither of the two preceding cases apply. The
existence of two half-lines (that are non-opposed) Pa and Ρ a* both
contained entirely within p involves the sane property for all half-lines Ρ a"
Interior to the angle aPi1 In the plane of this angle. The set of the half-
lines Ρ a In the interior of p thus constitutes a convex conical (or
pyramidal) volume, and all the points P. are In the Interior or on the surface
of the supplementary cone. The argument concludes as In the preceding cases.
68. If we were to construct about each point P., homologous to Ρ ,
the corresponding fundamental domain p., we would fill up the entire
Euclidean space without the different domains overlapping each other. All these
domains are equal to each other such that a kind of tessellation of the space
is obtained.
To the points P., which have effeotively served to construct th*
boundary of p, there corresponds a finite number of operations S, of the
-1 *
holonomy group. The operation S. , which takes P. to Ρ , takes Ρ to
tr tr О О
a point P. distinct from P.; otherwise the centre of Ρ P. would be Invarl-
r j I o t
ant under S. contrary to one of the fundamental properties of the group (see
No. 65).
To the points P. and P, there correspond two plane faces J. and 7.
t j t j
of the polyhedron p, and points of these two faces are pairwise homologous',
a point from J. by the operation ST - S,.
The faces of the fundamental domain p are thus pairwise homologous and
the operation uhioh brings about the ooin&idenoe of points of one with the
homologous points of the other, is that which takes the point Ρ to the point
P. symmetric to Ρ with respect to the second side.
These operations are said to be the generating operations of the holonomy
group, This title results by the fact that every operation in G results from
a composition, taken in the appropriate order, of the generating operations
carried out a sufficient number of times. Effectively, It Is possible to go
from Ρ to P. by crossing a certain number of fundamental polyhedra; the
emerging faces of successive polyhedra are homologous to the faces of p. It
suffices to carry out the generating operations associated with these faces,
and In the reverse order In which they are presented, In order that the
resulting operation takes p to p., that is to say Ρ to P.. On account of
this, the holonomy group thus admits a finite number of generating operations
by means of which the rest are generated. Incidentally, the holonomy group
contains an Infinite number of operations, otherwise the centre of the mean
distances fron Ρ and Its homologues which would be a finite In number, would
be Invariant under each of the operations of the group.

Locally Euclidean Riemannian Space
73
VI. DETERMINATION DF ALL NORMAL, LOCALLY EUCLIDEAN RIEMANNIAN SPACES
69. We are now In a position to relate the determination of all normal,
locally Euclidean Riemannian spaces to a problem In group theory.
Let В be a translation group (with or without symmetry) which has the
two properties mentioned In No. 65, I.e., It Is discontinuous and does not
admit any operation other than the Identity that leaves any point Invariant.
The Euclidean space, in which two points homologous with respect to the group
G are regarded aa identical and for which an ordinary metric Is adapted,
constitutes a normal, locally Euclidean Riemannian space. For our accounts, we
note that given any point Q In Euclidean space, all the homologous points,
by virtue of the discontinuity of the group G, are exterior points of a
certain sphere centred at Q and of radius R. Let us then take the sphere cen-
p
tred at Q and radius ·*. Two distinct points Q, and Q2, homologous to
each other, could not exist Inside this sphere as otherwise the operation In
G taking Q, and Q»· an operation which does not leave Q invariant,
would then take Q to a point Q' with
QjQ ■ Q2Q'
We would then have
QQ' * QQ2 + Q2Q'
or
QQ' * QQ2 + QQ1 < R ,
contrary to our hypothesis. Consequently every region of the Euclidean space
ρ
Inside this sphere of radius 7 Is formed by distinct points of Riemannian
space and the metric Is everywhere regular. The number r which corresponds
(see No. 57) to the point Q, regarded as a point In Riemannian space, Is
< R
It Is possible to have a more concrete representation of Riemannian space
by constructing a fundamental polyhedron which could be achieved by Just
knowing the given group G and taking a point PQ In the space and Its different
homologues.
The method that we have adopted to construct the fundamental polyhedron
Is not the only possible one. In some cases It Is an advantage to modify It.
Let us assume, for example, that the operations of the holonomy group G,
which leave Invariant the Euclidean metric, also leave Invariant another metric
defined In terms of rectangular co-ordinates by a quadratic differential form
with constant coefficients. It Is possible then to construct a fundamental
polyhedron by applying the method of ruling, but in this case, with this newly

74
Locally Euclidean Riemcamian Space
adapted metric. This new polyhedron will not be equivalent to the first, but
that Is not important.
If the Rlemannlan space Is closed* the different fundamental polyhedra
that мау be constructed are all bounded, and have the вате volume, I.e. the
total volume of the Riemannian space.
Ue add the remark that Riemannian spaces formed by a group S of
translations are strictly speaking, orientable\ the remainder being nonorientable.
VII. 2-DIMENSIONAL NORMAL, LOCALLY EUCLIDEAN SPACES
7D. Let us apply the preceding general principles to the case of two
dimensions. We have to determine all the discontinuous groups formed by
translations (displacements to be precise) or translations accompanied by a symmetry
with respect to a straight line parallel to the direction of the translation.
An orientable epaoe will thus give a holonomy group, uniquely formed by
translations. The set consisting of a point Ρ and Its homologues will then
form either a linear lattice or a plane lattice. The first case corresponds
to the cylinder, the second to the torus with a Euclidean metric. The first
spaces distinguish themselves uniquely from each other by the extent of the
generating translation. The remainder {Clifford apaoea)t by more essential
properties. To each of them there Is associated a system of elliptic functions
(doubly-periodic). It Is the rrodulua of these functions which essentially
differs, from one to the other. In the corresponding spaces.
The fundamental polygon could here be simply constructed by applying what
was said In No. 69. Every translation group effectively leaves Invariant all
the metrics with constant coefficients. If we then consider, for example, the
group
(x1 » χ + pa + qal
(4)
yl ■ у + pb + qbl
where ptq € Ζ arbitrary and ataltbtbl are constants, It will suffice to
consider the metric In the case where {a,b) and (α',ύ*) are unit,
rectangular vectors. The polygon obtained by the method of ruling Is a parallelogram
having two sides equipollent to the two vectors respectively.
71. The theory of analytic functions of a complex variable applies easily
to Euclidean metrics, at least In a prescribed domain. Let /(e) be an
analytic function, holomorphlc In a neighbourhood of ζ , and which does not vanish
for ζ ■ г . The equation
deZ= |/(β)ώ|2

Locally Euclidean Riemannian Space
75
defines, In the plane of the conplex variable «, a regular Euclidean metric
In a neighbourhood of я . If 6{я) Is a primitive function for /'(*)» e.g.
0(л) - / f{z)dM
/.:
a function defined In a neighbourhood of г , and If we put ф{я) в Р + £Q,
we have
2 2 2
ds ■ dr + dQT
This result suffices to show that the metric Is Euclidean. It Is, on the other
hand, regular since by putting
* * χ * iy* /(*) β Mx,y) + гЪ{х,у)
we have
ds2 - (A2 + B2)(da2 + dy2)
2 2
and the coefficient A +B does not vanish at я .
о
It Is easy to find locally Euclidean manifolds with an everywhere regular
metric. Firstly we take for /(*) a rational function of s, nonvanlshlng
for finite s:
/(■) " qJTJ- · /(■) * 0
where Q(«) Is a polynomial In я. For я Infinite, the metric da ■
stays regular If when putting * ■ τ» the Integral element χπ- takes the
form 6{t)dtt with ^(o) f 0. This Is the case If Q(«) Is a polynomial of
the second degree or more. In this case the manifold constituted by the plane
of the complex variable я (with the point at Infinity assumed) has an
everywhere regular metric, except for the two zeros of Q(s). If these points are
removed from the manifold, points which are at infinity with respect to the
given metric, a two-dimensional normal, locally Euclidean Rlemannlan space Is
obtained with boundaries at Infinity. It Is on the other hand, evidently
orlentable. Consequently It develops on the Euclidean plane as an infinity
of strips, as for a cylinder. On denoting by и and ν the rectangular
co-ordinates of the Euclidean plane, it may be seen that я ieji doubly-
periodic function of и + iv. The Inversion of the Integral
thus gives a uniform, simply-periodic function.
It Is also possible to consider a non-uniform function /(*); there
exists then a Rlemann surface on which It Is uniform. For example If we take
1
1 -*2
a.2
2
te a doubly-
/ dz
az + Ъя +1
/(*) ■ j ■ , the metric

76
Locally Euclidean Riemannian Space
da'
Is, as can be easily calculated, regular at all points of the Rlemann surface
except at the two points *■» on the two sheets. If these two points are
removed from the manifold, a manifold with two boundaries Is obtained but they
are situated at infinity with reepect to the metric in question. He would then
obtain by Inversion a simply-periodic function. Finally let us take an
elliptic Integral / JBffr where R(s) Is a polynomial of the fourth degree. The
metric
da'
dz
Щ*1
Is everywhere regular on the Rlenann surface attached to the function ЩГ) ,
and this Is without exception. A closed manifold with an everywhere regular
Euclidean metric Is thus defined. Consequently this manifold develops on the
Euclidean plane as a system of parallelograms and *, a uniform function of a
point of the nanifold, is an analytic uniform function, doubly-periodic in the
complex variable u + iv9 where и and u are the rectangular co-ordinates
of the Euclidean plane on which 1t develops. Ve have thus verified the
fundamental property relative to the inversion of the elliptic integral.
The same reasoning would apply without any modification to the Integral
of the first type attached to any algebraic curve of rank 1 and degree p.
f{*,t) = 0
The metric
, 2
OB ■
Q(»,t)d»
where Q Is the adjoint polynomial of degree
if/it
ρ - 3, Is everywhere regular on the Rlemann surface relative to the curve In
question.
Г da
This argument would not apply to a hyper-elliptlс integral ] vft-t ·
because the metric would cease to be regular at the two points л = «, these
being points which are not at infinity for this rntrio*
72. Let us move on to non-orlentable spaces. Every operation of the
group G having a symmetry could be defined In rectangular co-ordinates by
equations of the form
x" ■ χ + a
У1 " -У
(5)
This operation, repeated twice 1n succession, gives a translation parallel to
Or, of magnitude 2a. Thus there exists In G translations parallel to (b.

Looally Euolidean Riemcamian Space
77
Let α be the minimum (a > 0) extent of these translations. It Mould always
be possible to recall α as having a value less than a, and as 2a must be
a nultlple of a, 1t 1s possible to assume that α - £. We would then have
1n the group G, the operations
(x' ■ χ + na
(6)
(x' - χ + (n + »s)a
(7)
yl - -y .
Let us suppose that In G, there were translations not parallel to Or. Let
x' ■ χ + λ
у* ■ у + u
be one of the translations. G would then contain the translation
*■ - χ + λ
у1 ■ у - u
resulting In the three successive operations
x' ■ r + -j- ι x' ■ χ + λ , χ* ■ χ - -я- ,
у* - -у , у* - у + и , у" ■ -λ
Consequently we would also have the translation parallel to Or of extent
2λ. Thus λ Is an Integer multiple of | , a nultlple that could always be
reduced to 0 or j. Now 1t 1s not reducible to j, unless G contains
the operation
y* ■ -y * u .
which leaves Invariant the point (0Л). Thus G contains a translation
parallel to Oy. What results straight away Is the general form of the
operations of the group:
(x* ■ χ + pa
(8)
y' ■ у + qb

78
Locally Euolidean Ri^mannian Spaoe
■ x* ■ χ + (p + *s) a
•у' а -У + qb
(9)
where ρ and q denote arbitrary Integers.
73. There thus exists two classes of locally Euclidean nonorlentable
Rlemannian spaces corresponding to the two classes of or1entable spaces.
It Is possible to take for the fundamental domain of the spaces of the
second class a rectangle centred at the origin and whose sides are parallel to
the axes and of respective length j and b (Fig. 2).
The two operations generating* the group are
*♦!
x' - *
and
y = -у
у + Ъ
The first takes the side АО onto the side CB, the frame attached to a point
И of AD changes In sense at the homologous point H' on CB. The second
translates AB onto DC; the frame attached to Ρ on AB preserves Its
sense at the homologous point P1 on DC.
74. Let us dwell for a moment on the first class of nonorlentable spaces
riff.·.
whose holonomy group Is defined by the relations
rx' ■ χ + na
V ■ у
.χ' = χ + {η + 4)α
•V ■ -у
(б)
(7)

Looally ЕиоНаяап Riemcmnian Space
79
It Is possible to take as the fundamental domain (Fig. 3) the strip of the
plane taken between the straight lines x-0 and x-§ . and
ГЦ.».
Η
О -
the generating operation that takes the first straight line onto the second Is
.♦t
But alternatively, a better fundamental domain could be taken by replacing the
part of the first which 1s below Or by Its homologue relative to the
generating operation. We would thus obtain (Fig. 4) the region of the plane situated
above Or between the lines x-0 and χ-α.
*
nf. ι
* ■ V *
This domain corresponds to two generating operations. The first of them,
defined by
x' ■ χ + α
У' в У

80
Locally Euclidean Riemannian Space
takes the indefinite side x*0 to coincide with the Indefinite side x = a.
The second, defined by
*' = x + f
Ущ = -У
takes the side 0A to coincide with Its extension AB. The correspondence
between the points situated on the boundary and the frames attached there 1s
shown In the diagram.
The space admits a finite straight line of length |, I.e. 0A - AB.
The sides parallel to Ox are also finite but of length 2a; the sides
perpendicular to Or are Indefinite in both directions and are represented
Inside the fundamental domain by two half-lines running In the opposite
direction, the first ending for example at Q and the second leaving the homologous
point (Fig. 4). As for the other lines, they are also Indefinite In both
directions. The diagram (Fig. 5) Illustrates one, taken entirely within the
Interior of the fundamental polygon. It can be seen that it oroeees itself at an
Infinity of points; If It is followed fro» its starting point Ρ - the first
point where it crosses Itself is between PI and P,..
It then follows on 1n a regular fashion, which from the "analysis situs"
viewpoint, means that the line In question follows a pattern represented by the
diagram (Fig. 6).

Locally Euclidean Riemannian Spaoe
81
It may be seen In this example that In certain locally Euclidean Rlemannlan
spaces, direotion ie not absolute. A vector whose origin changes and which
stays throughout, parallel to itself, may return with a different Initial direс
tlon when the origin describes a cycle. In this particular space, there are
two absolute or fixed directions: they are the direction parallel to the χ
axis and the perpendicular direction.
75. The two classes of closed locally Euclidean Rlemannlan spaces both
have a parallelogram as their fundamental polygon. In both cases It may be
easily verified that there are two straight lines (not homologous to each
other) and a single vertex (the four vertices being homologous to each other).
On denoting by A the number of sides and S tie number of essentially
distinct vertices, we thus have
A - S + 1
It is possible to link this relation to a rather more general property. Let
us decompose by whatever means available, a two dimensional, closed (having a
finite total area) locally Euclidean Riemannlan space into a certain number
of sufficiently small polygons, to which may be applied the theorems of plane
geometry. Let F be the number of these polygons, A the number of their
distinct sides and S the number of vertices. The sum of the angles of a
polygon with η sides Is
π(η-2) .
Consequently the total sum of all the angles of all the polygons 1s
π(2Α - 2F) .
On the other hand, about each vertex the sum of the angles Is 2π and thus
we have the relation
π(2Α - 2F) = 2nS

82
Locally Euclidean Riemannian Spaoe
or
F + S - A .
In accordance with a theorem of "analysis situs," If this relation Is true for
a particular decomposition, then it Is true for any other decomposition,
particularly when the total space Is seen as forming a single polygon; we then have
F-l and A-S+l. This way, we might recognise a theorem that was proved
previously: the surface of a sphere cannot be endowed with an everywhere
regular Euclidean metric. Effectively, the decomposition of the surface of the
sphere Into polygons gives rise to the classical formula given by Euler, I.e.
F + S - A + 2
76. The determination of the discontinuous groups G of 3-dlmenslonal
space could be made without too much difficulty- If the Rlemannlan space Is
orlentable. Its group G would only contain translations and hell colda1
displacements. Apart from the case where G Is generated by a single hellcoldnl
displacement
x* » χ cos πα - у sin πα
у' ■ χ sin па - у COS па
а' ■ s + пп
where n Is an arbitrary Integer, h a given constant length, and α a
constant angle, the hell colda1 displacements which figure In G correspond to an
angle of rotation equal to Ϊ» Ϊ·, =γ or π. The fundamental domain In the
first case mentioned, where the angle α could be Incommensurable with n,
Is for example* the volume of the space contained between the two planes
s -0 and л ■ h.
VIII. NORMAL, LOCALLY EUCLIDEAN RIEHANNIAN SPACES AND ELEMENTARY GEOMETRY
77. In every sufficiently small region of a normal, locally Euclidean
Rlemannlan space, all the theorems of elementary geometry are tenable and It
Is possible to ascertain that one Is not In a Euclidean space providing there
Is departure outside this region. Certain axiom* which appeal to the
properties of the space taken In Its entirety remain true, but the others cease to
be so.
Amongst the first, let us Indicate one of them, I.e.: Through any two
points of the apace, there раееев a straight line. By representing a locally
Euclidean Rlemannlan space by means of Its fundamental polyhedron, the axiom
becomes evident.

Locally Euclidean Riemannian Space
83
Contrary to what Is expected, It Is only true for Euclidean space that
through any two points, there just passes one line. Let Ρ and Q be two
points of the fundamental polyhedron of a normal, locally Euclidean Rlemannlan
space. We represent the horoologues of Q In a finite number as Q,»(L,....
0 The sides PQ.PQi.PQ-,... »PQ ,... of the Euclidean space, brought
back If we wish. Inside the fundamental polyhedron, provide an Infinite number
of sides having the two points of the Rlemannlan space corresponding to Ρ
and Q.
It may be seen that the axiom, by which through any two points there
passes only a finite number of lines, would suffice to characterise Euclidean
space amongst all locally Euclidean Rlemannlan spaces.
78. We could take another point of view, as did W. Killing. In Euclidean
space, a movement Impinging upon a rigid body Is transmitted throughout to all
rigid bodies bound to the first. In a locally Euclidean Rlemannlan space, the
rigid bodies having sufficiently small dimensions admit the same degree of
mobility (without deformation) as In the Euclidean space. It Is no longer the
case If the body Is extensible; nore precisely. If It Is Impossible to contain
It within the Interior of a single fundamental polyhedron In Its development
In Euclidean space. The movement In a Euclidean space Impinged on the
neighbourhood of a point Pq, In Euclidean space results In a movement In the
neighbourhood of the point Ρ,, homologous to PQ, but this movenent. In
general, differs from the original one. The velocity of P, for example will
not be relative to the frame R, attached to P, as the speed of PQ was
to the frame Rf). It could be said that If a closed chain of rigid bodies Is
considered, each one Influenced by the preceding one, a movement Impinging on
the first one and transmitting by degrees would extend in Its return path to
Impinge on the first body with a movement quite different from the Initial
movement. In other words, locally Euclidean Rlemannian space does not aomlt
the complete mobility of Euclidean space.
If a collective movement Τ of the entire space Is required, It Is
necessary and sufficient that the displacements P.Q. undergone by the different
points P. homologous to a point Pq are all produced In the sane way with
respect to the corresponding franes R.. The successive displacements S.
and Τ which take PQ to Q. via the Intermediary of P., must therefore
be equivalent to the succession of the same displacements, but In the reverse
order, Τ and S£ which equlvalently take Pq to Q. via the Intermediary
of QQ.
The determination of the combined motions that can be made in a Riemannian
epaae thus amounts to finding displacements in Euclidean space that are
interchangeable iri-th each of the operations of the hotonomy group.
We can easily show that if a displacement, whether or not accompanied by
a symmetry, Is Interchangeable with all the translations; It reduces to a

84
Locally Euclidean JHemcmnian Space
translation Itself. From this, there exists no displacement (whether or not
accompanied by a symmetry) which Is Interchangeable with each "proper"
displacement. We thus arrive at the following theorem.
Theorem. If a normal locally Euclidean, Riemannian epaae hoe the exact
mobility of a Euclidean apace, it may be identified with a Euclidean space.
2-dlmenslonal Clifford spaces only admit translations, whereas the
cylinder admits translations as well as symmetries with respect to the two finite
straight lines containing It.

Chapter IV
RIEMANNIAN SPACE AND EUCLIDEAN TANGENT
AND OSCULATING SPACE
I. THE EUCLIDEAN TANGENT SPACE AT A POINT
79. Let us consider a Rlemannlan space defined as a manifold endowed with
an arbitrary line element
daZ*g..dJ*J . (1)
He shall assume that the right hand side Is a positive definite differential
form whose coefficients are continuous and adnlt contlnous first order partial
derivatives. Part of the considerations In this section will not necessitate
the existence of derivatives. The easiest way of determining the geometric
properties of this space consists of an Identification with Euclidean space
In any uay possible. He take as our first step, the Introduction of the
Euclidean metric tangent to a point of the given metric от the Euclidean space
tangent to a point of the given epaae.
He shall call the Euclidean metric tangent to the point A(u ,...,u")
with the given metric, the metric defined by an Euclidean line element
da2 - γ-. dJ-dJ (2)
constructed with the variables u ,...,u", and such that γ.. а g>. for
и ■ и . It can be shown that there Is an Infinity of Euclidean metrics,
tangent to the given point; It Is sufficient In this case to take γ.. - (σ..) .
On the other hand, the set of these metrics Is Independent of the choice of
co-ordinates which analytically define the Rlemannlan space, for If the varl-
i i
ables и are replaced by some other determined variables ν , the new values
at A of the coefficients of the line element only depend on their original
values at A. The equality γ.. β д.. therefore holds for the new variables
at A.
Rather than to say that the manifold In question has been given a new
metric (Euclidean), It could be said that a representation of a Rlemannlan
space Is made on the Euclidean space, such that by this representation, the
2
line element of the Euclidean space becomes da . This Euclidean space will
be known as the Euclidean tangent space at A to the given Riemannian space\
this being convenient terminology to expound matters. It may be said that
there Is an Infinity of Euclidean tangent spaces at A; In this sense, the
2
line element do depends on an Infinity of arbitrary elements. But as we
85

86
Ов aula ting Spaae
are going to consider In the sequel, namely the geometrical properties common
to all these Euclidean spaces, we may quite conveniently say, the Euolidean
tangent apace at A.
80. An elementary geometrical concept, characteristic of the Euclidean
tangent space. Is that of the distance froa a point A to a point
Infinitesimal ly near, with a distance equal to da or perhaps da. It Is the same
concept which basically defines a Rlemannlan space.
The angle of the tuo directions d and б stemming from A Is given
In the Euclidean tangent space (see No. 29) by the relationship
γ.. du'uJ
cos Φ - г^— --
/ γ. rfu die / γ., бц 6ιΓ
For any Rlemannlan space, we can thus define the cosine of the angle between
the two directions by the formula
д.. dJtJ
cos * a i ■ / (3)
/ g.. du a\t / g.. ou Su
where the right hand aide σαη be eeen to be independent of the ohoioe of
oo-ordinatee.
Likewise the angle of a p hyper-plane with that of а ц hyper-plane may
be defined, and all the theorems of elementary geometry concerning angles all
from the вате vertex that are described between the lines, surfaces, etc.,
passing through this common vertex are valid for any Rlemannlan space.
81. At the point A of the Euclidean tangent space a veotor with contra-
variant components X1 (or covarlant X.) may be defined relative to the
natural frame attached to A (a frame that Is well-defined In Its extent and
form). At each point of Rlemannlan space. It Is then possible to define both
contravarlant and covarlant vectors. More simply, a vector may be represented
by considering a point In motion and the velooity of this point. Intrinsically,
It exists independent of the choice of co-ordinates In the Euclidean tangent
space.
Similarly, at a point A In Rlemannlan space. It Is possible to define
Ыvectors, multlvectors and a general collection of tensors. But the Idea of
the Euclidean tangent space Introduces In Rlemannlan Geometry particular
properties of each line, surface and volume of these spaces.
Firstly, given the elementary distance between the two points, the arc
lengths of any curve will be obtained by Integration. I.e.,

Oeoulating Space
87
■ feij *****
From this It would be Immediately possible to deduce, as did Rlemann, the Idea
of a straight line or geodesic as giving an extreme for the distance. We
merely Indicate at this stage, this possibility, and return to It later.
The elementary volume of the space, on account of the expression for the
elementary volume of the Euclidean tangent space. Is given by
di * Jq dudu- ... dun . (4)
From this a finite volume will be defined by the Integral
du du ... du
ff^l **
It Is thus possible to determine the area of a region of a surface. In three
dimensions. If the co-ordinates of a point on the surface are defined as
functions of two parameters α and β, a surface element stemming from A Is
а Ыvector whose contravarlant components are
2 3
*^*^ " Ρ£(α!%)^ **** " *23 <**** ·
dJdu* - Pftk|jj)* ArfB ■ α31 dad» .
The magnitude of this surface element Is thus
/я—7~3i—T~T2—а M
a23 a 31 a a12 "°"P
From this, the expression for the area of a finite region of the surface
be deduced.
//■
/23 Γ"31 ΓΪ2— a j*
82. Given a vector field In Rlemannlan space» It Is not yet possible to
know how to define the tenaor derivative of this field. Nevertheless Its ourl
may be easily defined via the Pfafflan
*
We have already utilised this concept In defining In No. 55 the distance AB
between two points In Rlemannlan space.

Oeoulating Spaoe
tydu + Xtfht2 + ... + X dun
which has an intrinsic significance in the Euclidean tangent space at each
point.
The bilinear covariant
ι /ЭХ. ЭХ ·\ · . · .
i (-* - -4) (AW - ^бцг)
leads straight away to the tensor
ЭХ. ЭХ.
χ.. - -Л —\ .
which 1s the curl.
The elementary vector flux which equally occurs in the Euclidean tangent
space, I.e.
/y (X du du + X du du + X Л* Л* ) *
leads, via the application of Ostrogradsky's f omul a, to the divergence (see
No. 44)
d1v χ ■ —
■G
a(,ff x1) , a</7 x2) , a(*gx3)
Эк Эк Эй*"
(5)
Similarly, given a scalar field V(J un), the two Beltrami differential
parameters (see No. 42) are defined without difficulty
83. Despite the vast number of concepts of Euclidean origin that we
could have generalised in any Riemannian space, it 1s the elementary
fundamental concepts which again warrant our attention. As an example, the angle
between two directions steaming froa two distinct points. Broadly speaking,
for each geometrical concept arising at each point, a scalar 1s readily
generalised. It 1s the same concept that occurs with one or several vectors, but
itith the condition that they all originate in the вате way. The divergence of
a vector field may be seen to be an exception, but that 1s only because we
have been able to deduce the Idea of the elementary vector flux, uhiah in

Osculating Space
89
practice only occurs at a single point in the field. On reviewing what we
have at the moment. Riemannian space 1s for us a collection of small fragments
of Euclidean space» but remains to an extent amorphous, because we have not
yet connected these various fragments by defining their mutual orientation.
This 1s what we Intend studying with regards to the osculating Euclidean space.
II. THE OSCULATING EUCLIDEAN SPACE
84. An Euclidean metric osculating at A(uM with the given metric will
be defined by a line element
da ■ γ.. du du
such that Its coefficients, along vith their firet order partial derivativest
have the same numerical values at A as the given line element.
Given there exists osculating Euclidean metrics, the set of these metrics
1s Independent of the choice of co-ordinates, since for a change in the given
variables* the new values of the g, . and their first order partial
derivatives are continuous since the original values of these same quantities were
known.
Rather than speak of an osculating Euclidean metric, we would rather say
an osculating Euclidean space.
The first thing to do 1s to prove that osculating Euclidean spaces exist
at a given point A of Riemannian space.
ь
Now if such a space exists» the coefficients Г.. which allow us to
w
localise the natural frame attached to the point \иг + du1) with respect to
the natural frame at u% are given by the relations that we established in
Chapter II (No. 33).
It thus suffices for us to show that it 1s always possible to find
coordinates (иг) in Euclidean space, such that at a point 0 in this space,
the д.. and the v.. have some given numerical values (knowing the д.. and
—ч- gives the Г.. and vice-versa). Now the Euclidean relations
Ъи% tJ
dH = du ek
yield
de. « ή^\
ЭН _ Э2Н а Jc
V'1 ЪиЧЛ Ч°к
Having established this, let us take at the point 0, a Cartesian frame

90
Osculating Space
(«,,...,«) defined by the numerical values of the coefficients д.. at A,
and determine the co-ordinates χ (with respect to this frame) of a variable
point И
in this space» with the condition that for u ■
3M -. ?*. - Μ ) „
*Λ '* · bJ ъЛ \Ъ) ** ■
'V
This is possible in an infinity of ways. For exempts it suffices to take
χ
О
The line element of the Euclidean space, relative to this system of
curvilinear co-ordinates, will be seen to have, for иг * цг, it coefficients and
о
their first order partial derivatives having the same nunerical values as for
the line element of the given Riemannian space. This 1s what was to be'proved.
85. All the geometrical properties common to the different osculating
Euclidean spaces at a point are seen to be the Intrinsic geometrical
properties of the Riemannian space. These arise from determining only the numerical
values at A of the y.. and ΓΪ..
Let us Imagine that in the osculating Euclidean space, the point A 1s
fixed as well as the natural frame (R ) attached to it. Each point Μ in
Riemannian space, 1s represented by a point Μ whose co-ordinates, relative
to (R ), are well defined to irithin an order of infinitesimals greater than
the second. The natural frame attached to the point R in Euclidean space,
with respect to Its magnitude and orientation 1s well defined to within some
order of infinitesimals greater than the first\ moreover 1t 1s equal to the
natural frame attached to Μ in Riemannian space, to the same order. Every
vector of the Riemannian space attached to Μ 1s represented with the same
degree of approximation by an equal vector. In general, if И and N are
any two points (neighbouring A) 1n the Riemannian space and if their
representative points И and ff in the osculating Euclidean space are contained
Inside a sphere centred at A and of radius r, the scalar products of the
vectors representing two vectors attached to Μ and N respectively are well
defined to within an Infinitesimal relative to r.
A vector χ attached to A and a vector r' attached to an 1nf1n1tes1-
mally close point A' in the Riemannian space are represented in the
osculating Euclidean space by two vectors whose geometric difference is» as a result
of what was just discussed, well defined to within an order of Infinitesimals
greater than the first.
This leads us to the extension to any Riemannian space in terms of the
concept of abaolute or aovcoriant differential of a vector, or more generally,
of a tensor. We shall continue to denote this differential by D.

Oaculating Space
91
By definition, we will then obtain,
DX£ - dXf + Χ* ω* (8)
DX. - dX. - X. u* . (9)
In the meantime it 1s Important to note here, that the absolute differential
in Euclidean geometry was a true differential, whilst this 1s no longer the
case in Riemannlan geometry.
In particular, by denoting by е., we have the vectors with co-ordinates,
De. - υ* e, .
г г к
Two vectors attached to the Infinitesimally close points A and A' will be
said to be equipollent if they are equipollent in the osculating Euclidean
space at A» or, if the absolute differential of the first vector (when this
first vector 1s taken to the second) 1s zero; this amounts to the same thing.
Thus the conditions for equipollence are
dX£ + X* w\ » 0 (10)
dX£ - Xk w\ « 0 . (11)
To transport by equipollence a vector χ attached to A to a point A'
Infinitesimally near, 1s to construct the vector x' attached to A',
equipollent to x. Clearly this vector 1s only defined to within an order of
Infinitesimals of order greater than the first. This operation, in the
osculating Euclidean space at A, 1s represented by a simple translation from A to
A'.
Transport by equipollence may be seen to have some Important properties:
if two veotore are transported by equipollence from their oonrnon origin A
to A't their eoalar product ie invariant.
If one wished to prove this theorem by analysis, it would suffice to
remark that the differential calculus of the scalar product would only Involve
the numerical values at A of the д.. and their first order partial
derivatives. The proof thus made would be trie same if the space were Eudlldeon, in
which case the theorem 1s quite clear.
The absolute differentiation will be applied to any tensor field. In
particular, the fundamental tensor д.. has Its absolute differential
Identically zero.
Lastly, the tensor derivatives are defined as in No. 41.
86. The definition of the acceleration of a moving point in Riemannlan
space now comes about quite easily. If ν 1s the velocity vector, the

92 Oeoulating Space
acceleration vector 1s ^ . Similarly, accelerations of different orders
could be defined. The contravarlant components of acceleration are» as in
Euclidean geometry,
au , pi du du
and Its covarlant components are again determined by Lagrange's algorithm.
A moving point whose acceleration 1s zero throughout always has constant
speed and 1s parallel to Itself. More generally a "straight" line of Rleman-
nlan space will be a curve whose tangent stays approximately parallel to it.
If such a curve 1s traced out by a variable whose velocity 1s unity (1), the
acceleration of the variable 1s zero.
The equations
dV+r£ *£. a£. 0 (12)
de kh ds Ж
thus defines the "straight lima" (geodesies) of Riemannian spaces. Later (No.
95) we shall see that the straight lines of Riemannian space are curves of
least distance; they are thus seen to be equivalent to the Riemannian geodesies.
The dynamical equations of the point readily generalise from Euclidean to
any Riemannian space; they may be written in one of the two forms
γ * F
ч * h ·
depending on whether the contravarlant or covarlant components of the force
are used.
It 1s worth noting how the general dynamical equations of the systems
(holonomic with constraints) reduce to the dynamical equations of the point,
by regarding the locus of a rigid system as a manifold in which each element
(or point) woild be one of the positions of the system. On this nanifold, by
adopting the product of the effective force of the system and the square of
dt as the line element, the covarlant components of the acceleration of a
point on the manifold are (see No. 37)
d ЭТ ЭТ n,*
ч = ш 77-27-77 ■ (13>
Following Lagrange's Equations, the most general form for the motion of the
system is given by the equation
*i " Q£ '

Osculating Spaoe
93
where Q. 6иг denotes the Infinitesimal work done by the given forces. There
ie thus a perfect oorrespondenae between the motion of the given eyetamt
assumed to have η degrees of freedom, and the motion of a point in n-dimen-
eional Riemarmian spaoe. The correspondence preserves the effective force as
well as the sum of the elementary work of the forces.
87. The theory of curvature extends without modification from Euclidean
to any R1emann1an space. Let Μ be a point on the curve, t the unit
vector tangential to the curve at M, da the arc element of the curve. The
vector -jj- 1s normal to t (this may be clearly seen by referring the osculating
Euclidean space at M).
Let
ρ
Ъш -"
where η denotes a unit vector (principal normal). Let Ь be the unit vector
perpendicular to t and n, forming an exact trihedral, and also let
^ - at + βη + yb
j§- a't + 6'n + Y'b .
The relations
2 2
t · η ■ 0, t · b ■ 0, η · b ■ 0, η - 1, b - 1
on absolute differentiation, yield
j- + α - 0, α' - 0, γ + β* = 0, 6-0, γ' - 0 .
By then putting
γ β τ .
the generalised Frenet formulae are obtained
*
For properties arising from curves and surfaces in ordinary space, see
Elements de Geometrie infiniteaimalet by G. Julia (Paris, 6auth1ers-V1liars
1927).

94 Osculating Space
Dt . 1
■3- a - η
de ρ
jfe. -lt+lb (14)
Ob 1
Α Τ" *
The quantities - and - are known respectively as the curvature and toreion
of the curve.
The "straight" lines of Riemannian space are curves of zero curvature.
As for the curves of zero torsion, they could be characterised by the property
that the osculating plane element at a point Μ of the curve 1s parallel to
the osculating plane element at a point M' infinitesimlly near. The vectors
t + Dt, η + Όη which are the unit vectors taken along the tangent and
principal normal at M', effectively become transported by parallelism free H1
to M,
. . de ώ , , ώ , .
t + — η η fc + — Ь »
Ρ ρ τ
they occur in the osculating plane element at Μ only in the case when the
torsion 1s zero.
If the Riemannian space 1s represented on the osculating Euclidean space
at M, the given curve (C) has for Its Image a certain curve (Γ) which
hae the вате curvature at Μ ав (С). Moreover the vectors t, nt b on (Г)
are the Images of their analogues on (C). This 1s actually seen ноге clearly
once represented in the following way.
When in Riemannian space, the given metric en every osculating Sue Η dean
metric at Μ ie adapted, the curve (C) has at Μ the вате tangent,
principal normal, binortnal and curvature; but it does not neoeeearily have the
лоте toreion vith the two me trice. In particular every straight line in
Riemannian space presents a point of Inflexion at Μ for an observer who would
adapt the osculating Euclidean metric at M.
88. In the light of the classical theory of the curvature of surfaces,
the above discussion extends quite easily to Riemannian space. The different
curves traced across a surface (S) and passing through a given point Μ on
this surface have the same principal normal and curvature» in accordance with
the Riemannian or the osculating Euclidean metric adapted to И. It so happens
that the laws which govern the variation of the curvature of these curves when
their tangent at Μ 1s varied are the sama as for Euclidean space. Every
curve tangent to another has the same normal curvature. This normal curvature
1s equal to co* , where V denotes the angle between the principal normal
to the curve with the normal to the surface and ρ the radius of curvature of
the curve: Thie ie Meuenier*e theorem.

Oeculating Space
95
There exists two rectangular directions tangent to the surface
corresponding to the maximum and the minimum normal curvature; these are the principal
directions with the corresponding normal curvatures called the principal aurva-
turee* The lines of curvature are the lines which, at each of their points,
are tangent to one of the principal directions at that point. If θ denotes
the angle made by one curve with the line of curvature of the first family,
we have for this curve
cos V _ cos2 6 , sin2 θ ^ (15)
where we denote by J- and -5— the two principal curvatures.
Rl K2
The aeynptotio curve в are those whose normal curvature 1s zero; they are
characterised by the property that their osculating plane element 1s tangent
to the surface. They could be characterised again by the condition that the
straight line of the Riemannian space tangent to Μ with an asymptotic curve
has second order contact at this point with the surface. This amounts to
saying that in a Euclidean space, a line tangent to a surface at a point M,
admitting at Μ a point of Inflexion, has second order contact with this
surface when it 1s tangent at Μ to one of the asymptotic tangents of the
surface.
We can also characterise the lines of curvature by the condition that
for any given point Μ on the curve, the normal to the surface at a point M*
on the curve Infinitesimally near to M, once transported by parallelism from
M* to M, occurs in the same plane element with the normal at Μ to the
surface and the tangent at Μ to the line.
If the unit vector normal to the surface 1s denoted by ν and the unit
vectors tangent to the surface by t, and t?t we have by a displacement in
the direction of the vector t. with length de,
da-,
°"-"ΐς-*ι ·
and by a displacement in the direction of t~ with length de~t we have
These relations represent the classical formulae due to Olinde Rodrigues. If
a displacement de 1s made in a direction making an angle θ with t,t then
in general we have
Dv B cos 6 , sin 6 . nc\
ш —iq- *i " -R- c2 (16)

96
Osculating Spaa*
The total curvature
1
о- of the surface at a point Μ could finally be
defined by the approach taken by Gauss for a Euclidean space. Let us consider
a surface element da enclosing the point Η and, in the Euclidean tangent
space at M, the sphere of centre Μ and of radius 1. Let us transport by
parallelism at Μ the normals to the surface at the different points of the
element do, and consider a small region of the sphere determined by the
tracing out of the normals so transported. The ratio ^ of the area of this
portion of the sphere to the given area, 1s equal, in Its maximum, to the total
curvature ^-g- . We could then regard Δ> as the solid angle of the cone
with vertex Μ obtained by transporting by parallelism at Μ the normals at
the different points of the element da.
89. We can extend to the more general R1emann1an spaces the concept
of geodeeio toreion of a curved trace on a surface. For this purpose, let us
denote by tt η and b the unit vectors along the tangent, principal normal
and binormal to the curve respectively, and continue to denote by t,, t~ and
ν the unit vectors along the principal tangents and the normal to the surface.
Finally, let θ be the angle made by the tangent to the curve with the first
principal tangent and V the angle of the principal normal to the curve with
the normal to the surface. The direction cosines of the vectors t> пл b
with respect to the vectors t., t_ and ν are given by the following table:
t
η
b
Ί
cos θ
- sin θ sin V
sin θ cos V
h
sin θ
cos θ sin V
- cos θ cos V
V
0
cos V
sin V
Let us start with the generalised Frenet formulae
Dt _ 1
-τ· - — П
da ρ
Db_ 1
as" "T"
Π4)
as well as the formula
Dv _ cos θ
За-
Ч
Ί "
sin θ ,
[16)

Osculating Space
97
If we derive the relations
t · и ■ 0 , и · о ■ cos ν , ft · у я sin V ,
we obtain two distinct equations. I.e.
cos v _ cos2 e sin2 e
ρ R1 R^ CIS)
f + T-(^-l^) Ч" · cos θ · (17)
The second of these equations has on the left-hand side the geodesic torsion
of the curve. The form of the right-hand side shows that It Is the same for
all curves which have the same tangent, and It Is zero for lines of curvature.
When applied to the case of a distinct asymptotic line of a straight line
(V - J), relation (17) gives
T-df-iq) sinecose ·
Now for an asymptotic line.
2 2
cos e sin e n
в в
Kl κ2
and we deduce that
(10)
For any Rlemannlan space, this Is the Beltrami-Enneper theorem.
Theorem. The asymptotic linee stemming from a point Μ of a surface have at
that point equal and opposite torsionsM whose magnitude ie equal to the square
root of the total curvature of the surface with the sign changed.
In particular the torsion of a iowiZy-asymptotlc line Is constantly zero.
It Is known that,1 η Euclidean space. If two families of asymptotes to a
surface are Identified, then the asymptotic lines are straight lines. This
theorem Is no longer true for an arbitrary Rlemannlan space. Nevertheless
It may be affirmed. In accordance with Enneper's theorem, that they are curves
with zero torsion.
E. Cartan. Sur lee oourbes de tore-Con nulle et lee surfaces developpables dans
lee epaoe de Fiemann (Comptes randus, t. 184, 1927, pp. 138-141).

98
Osculating Space
90. The theory of conjugate tangents generalises In any Rlemannlan
space. If Μ and M' are two points Inflnlteslmally close to each other on
a curve (C) traced on a surface (S), the plane element tangent to the
surface at M', being transported by parallellsn at M, has a certain direction
conjugate to the direction of the curve (C) at M, In comwon with the
plane element tangential to M. Two conjugate directions are harmonic
conjugates with respect to the asymptotic directions. Finally, the absolute
differential of the unit vector normal to the surface when we pass from N to N'
Is perpendicular to the conjugate direction of the curve. All these properties
result from the representation of the osculating Euclidean space at M.
91. Let us end by generalising a well known theorem by Oupln. A triply
orthogonal system Is commonly known as the system formed by three families of
one parameter surfaces that intersect at right angles. Dupln's theorem says
that the Intersecting curve of the two surfaces belonging to two different
families, Is a line of curvature for each of these surfaces.
The analytical proof that we are going to make will apply just as well
to Rlemannlan spaces as It does for Euclidean spaces. Let us take for co-
I 2 3
ordlnates, the parameters и , и , u , of the surfaces of the three families.
The spatial line element will then be of the form
2 2 2
d*Z ' gu idJ) *gn{d*Z) * 033 (<&3)
since the cosine of the angle between the two Intersecting curves with
different co-ordinates 1s zero. The condition that expresses, for exanple, haw the
co-ordinate line (С,) (и a variable) Is a line of curvature for the surface
3 1
5. (u a constant), Is that the vector «3 + Di«3<& , normal to this
surface at the point M' Inflnlteslmally near to Μ on (C.), Is In the same
plane element with the vectors «- and e,. In other words, the coefficient
of e? In the expression for Ο,β- Is zero. Now we have
°*3 * ω3 β2 + ω3β2+ω3 β3
°le3 " Jl «1 * Jl e2 + J? e3 ·
It 1s thus necessary to prove that ГТ ■ 0; or, as
31
2
Г321 * ff22 gl
It Is necessary to show that the Chris toff el symbol [32 1] Is zero. Now
this is clearly the case as g22 ■ g^ - g^2 = 0.

Osculating Space
99
III. THE OSCULATING EUCLIDEAN .SPACE TO A CURVE
92. It Is possible to obtain new geometrical properties and new theorems
by developing a curve In Rlemannlan space In Euclidean space.
Let us start from a point A on the curve and assume that the
co-ordinates of a point on this curve are expressed as a function of a parameter t
that Is zero at A . In a Euclidean space, let us take an origin to which
we shall attach a Cartesian frame (R ) determined In expanse and form by the
numerical values of the д.. at A . We shall let a point M* In Euclidean
°гс о
space and a Cartesian frame («.',...,« ') attached to M' correspond to
ι η
each point Μ on the curve with parameter t. We shall start, as we have
done already (see No. 46), with the differential equations
<2M' ■ du e.
. * (19)
where t Is the Independent variable. These unknown quantities Μ*, «'. are
determined by the following Initial conditions: for i» 0, the point M' Is
at 0, and the vector e£ is equal to the vector (e^) of the frame (R ).
It will be shown, as before (see No. 47), that the frame (R) attached to the
variable point t has the scalar oroducts of the vectors which determined It
equal to the coefficients of the corresponding g... Moreover, If Μ and
M, are two points Infinitesimally apart, whatever the given curve, and M*
and Mi, the corresponding points In Euclidean space, then two equipollent
vectors with origins Μ and H, have for their Images two equipollent
vectors (In the ordinary sense of the word) with origins M' and Mi.
Thanks to defining a Cartesian frame (R) at each point Μ', we have
In fact not only developed the given curve In Euclidean space, but actually
all of the Infinitesimal region of the Rlemannlan space about this curve. To
have the absolute geometrical variation of a vector, the origin of which
describes the arc of the given line In Rlemannlan space, it suffices to con-
etruot the ordinary geometrical difference between the too vectors obtained
in the preceding development. We also have a rigorous and precise evaluation
of this absolute geometrical variation, as well as knowing precisely what Is
meant by transport by equlpollence of a vector along the given path.
93. We would be more precise about the preceding operation by showing
that there Is an osculating Euclidean metric to the given metric along the
length of a given line, that there Is a Euclidean metric or even a Euclidean
space attached to this line. This means that It Is possible to determine a
Euclidean line element constructed with variables ub and such that, right

100
Osculating Space
along the line, the coefficients д.. and their first order partial derlva-
tlves have the same numerical values as for the given line element.
To prove this proposition, we shall consider the development which evolves
from the given curve In Euclidean space. To simplify natters, we are going to
assume that n = 3, and without loss of generality, that the line Is defined
1 2
by the equations и - 0, и а 0. The development has given us a function of
3
a point M' and vectors «i, β
In (19). we have
Γ e2f β3'
and on account on the equations
del
• du
(20)
t^
This established, let us determine as a function of и
2 3
иt и , a point Ρ
In Euclidean space, In accordance with the following Initial conditions: We
1 2
have, for и ■ и - 0,
ГР _ - -* .
Э(и ) 1 *
Э2Р
Г?" Й ·*
Э(«с)ь 2'
3 Р я г* „·
Эи Эй 1
(21)
where the ΓΪ are replaced by the numerical values that they take when we set
1 2 J-
и в и =0.
These conditions nay be seen to be compatible with each other ι It suffices,
for example to take
Ρ ■= M'+ ue\ + Aj + %(J)2 rf e£ + uu2 γ£ *£ + Ц(и2)2 г£ *£
fli/ fcfcte гв£л£ют, Euclidean еразв ίβ related to a system of aurvCtinaar oo-
12 3 12
ordinate в и л и , и . For и ■ и = 0, the natural frame attached to Ρ
(which reduces here to M') Is defined by the vectors
here to в
ЭР
5ΰ$
, which reduce
The coefficients of de of the Euclidean space are thus
the д., of the given da of the Rlemannlan space. As for the coefficients
This theorem was first proposed by Feral (Rend. Ace. Llncel, 31, pp. 21-23,
51-52).

Osculating Space 101
г. of the Euclidean space» they are obtained by taking the coefficient of
i? 2
el In ? Pi . If the Indices t and J are both distinct from 3, the
* ЪиЬЪий
coefficients themselves, In the Rlemannlan space, are determined from (21).
1 2
If i differs from 3 and j equals 3, we have for и - и - 0, on account
of (20),
/ Э2Р N я d (ЪР\ udBi ж Jc .
хЭи Эй '0 du N3u Ό du t
and the conclusion Is the same.
If finally, t ' j я 3, we have
Г Э2Р Ί ш <**3 я Jc. .
Г? 7Ϊ ГЛ '
.э("3) J0 л 3
In both cases the conclusion Is the same. The theorem Is thus proved.
It may be seen that the determination of an oeculating Euclidean space
along a given line ie made without integration, once the development
of the curve on the Euclidean space ie carried out.
94. The numerous consequences of the preceding theorem are very Important.
It may be seen In the first Instance, that an observer who movee only along the
given lines and is content to restrict his measurements to an itmediate neigh"
bourhood of this line . . . would not be able to see that he is outside a
Euclidean space for as long as he neglects infinitesimals of an order greater
than the first. Another Important consequence Is as follows: If In a
Rlemannlan space, an arc of a line Is considered Infinitesimally near to the given
line (C), the length of this arct measured by the Riemannian or the Euclidean
metric . . . is the same up to infinitesimals the second order.
In fact, at a point Infinitesimally near to the line (C), the
coefficients of the given line element and of the adjoining Euclidean line element
are equal up to Infinitesimals of the second order. The representation of
Rlemannlan space upon the adjoining Euclidean space thus preserves the distances
measured in the neighbourhood of the given line up to the same order.
95. When a curve Is developed on Euclidean space, the curve so developed
Is seen to have the same curvature and torsion at each point as for the given
curve, since In the development the absolute differential of a vector whose
*
It Is necessary to assume that the two arcs of the lines correspond, point
by point. In a way that the directions of the tangent elements corresponding to
the two points are Infinitesimally apart.

102
Osculating Spaa*
origin describes the curve, becomes the ordinary geometrical variation of this
vector. The generalised Frenet formulae thus become In the osculating Euclidean
space, the usual Frenet formulae:
A . 1 „ dn 1,J , db 1
da ρ · da ρ τ · as τ
A curve with zero torsion Is so developed along a plane curve, a curve
with zero curvature, along a straight line. The straight tinea of Hemannian
apace are therefore the curves whose development is along a straight line.
The Immediate result of this Is the equivalence of the definition of "straight"
lines given In No. 75 (which we adopted In the preceding case) and the
definition of the geodesies as given by Rlemann. Effectively, a curve (C) Is
developed along a straight line and In Rlemannlan space, a curve С Is
traced infinitesimally near, which stems from a given point A and abuts at
a given point В of (C), then the Image curve In the adjoining Euclidean
space will have the same length as (C) up to Infinitesimals of the second
order. Now 1л this Euclidean space, the Image curve of (C)' results from
the variation of a straight Иле segment; Its length Is thus the same as the
line segment up to Infinitesimals of the second order. Consequently. In the
Rlemannlan space, the first variation of the length of a line segment Is
Identically zero and clearly the "straight line" gives rise to the extreme* of the
distance conforming to the classical definition as given by Rlemann.
Another consequence arises; If a geodesic arc AB and an Inflnlteslmally
close geodesic arc A'3' are considered, then we have, as in Euclidean apace,
up to infinitesimals of the second order
arc A'B' - arc AB - -AA' cos(AB.AA') - BB' cos(BA,BB')
In particular, let us take a constant length R starting from A, along the
different geodesies stemming from A. We obtain a certain surface similar to
the sphere as a region between the extremities. The tangent plane element to
this surface at a point Μ is normal to the geodesic stemming from A which
abuts at M. The same applies If a family of geodesies generating a surface
stems from A only. If a constant length Is taken on each of them, the line
element tangent to any point И In the region between the extremities Is
normal to the geodesic stemming from A which abuts at M.
One other consequence Is the generalisation of properties of parallel
surfaces. If the normal geodesic Is constructed at different points of a surface
and a constant length Is taken on this geodesic, the region of the points so
obtained Is a surface which, at each of Its points, Is normal to the
corresponding geodesic. The result Is that If one two-parameter family of geodesies Is
normal to a surface, It Is normal to an Infinity Of surfaces.

Oeculating Space
103
96. The consideration of the osculating Euclidean space lends Itself to
generalising certain classical theorems quite easily; an example Is
JoaahimBtal'v Theorem. We shall consider a line (C) and two surfaces (S,)
and (Sj passing through (C). If (C) Is a line of curvature for each of
these surfaces, this property Is retained under the development of the attached
Euclidean space. Consequently, In this adjoining space, the two surfaces will
Intersect at a constant angle; it will therefore be the earn in the Riemarmian
epaoe. The converse Is just as evident.
97. What we are going to establish for a line could not In general be
possible for a surface. Thie does not exiet in general for a Euclidean epaoe
attached along a eitrfaae. The reason Is quite Simple, In so far that It Is
not possible In general to develop a surface on Euclidean space. To simplify
matters, let us assume, without loss of generality, that the surface Is defined
by и ■ 0. The development would require Integrating the total differential
1 2
equations In two Independent variables и and и
(<m - dJej + du2el2
(22)
ав^ - (ή du1 + ndu2) «' ;
i i
but the compatibility conditions are not In general fulfilled.
There Is nevertheless a retalnment of an Important result, if the
development on Euclidean epaoe ie poeeible, there exiet» an attached Euclidean epaoe.
In fact, to say that the development Is possible is to say that there exists
In the Euclidean space a point M* and vectors β,, el, e' as functions of
12 ι £ л
и and и satisfying equations (22) and the relations
.;..;.,„ (J.»2.o) .
This established, let us determine In the Euclidean space, a point Ρ being
12 3 3
a function of и , u , и by the condition that for и «0
Ρ - Μ'
a(uJ)'
This Is possible In an Infinity of ways. It suffices for example, to take
Ρ ■ Η' + Λ} + %<*3)ΖΓ3* шшк .

104 Qeaulating Space
12 1
This also relates to Euclidean space with curvilinear co-ordinates и , и , и .
For и ■ 0, the natural frame defined by these curvilinear co-ordinates Is
precisely
ЭР ш ЭМ' . ЭР ж ЭМ' . ЭР я ,
Эй Эй Эй Эй Эй
The coefficients of the spatial line element are thus the same as those of
Rlemannlan space, for и - 0. As for the coefficients ly., they are determined,
о *J
3 IP
for и = 0, by the vectors ■ ■ . Now we have for i,j f 3,
9u bit
2 2
ЭР Э Η'
г*
βι
Эй1*/7 ЭигЭи^ ^ *
for t * 3, j - 3
э2р , ,,„4 a.i
Эй Эй
Finally, for i ■ j ■ 3
Э2Р . Jt ,
7? 'зЛ
This demonstrates the equality, for и - 0, of the partial derivatives of the
2 2
coefficients of the ds for Rlemannlan space and the de for Euclidean
space.
We might remark that the possibility of the development of a surface on
Euclidean space Is assured If the transport by parallelism of any vector across
the surface Is holoncmic* that Is to say, It gives a result Independent of the
path taken on the surface In going from the origin of the Initial vector to the
final vector.
98. We might mention a final application of the Idea of the joining
Euclidean space. Let us consider a surface (S) In a 3-dlmenslonal Rlemannlan
space and a geodeeic (C) on this surface, that Is to say a line which Is
actually the minimum distance on the surface. In the attached Euclidean space
along (C), the line (C) moreover has a stationary length relative to all
the curves traced across the surface with Its extremities at two given points
of (C). Consequently, by virtue of the classical properties of geodesies of
a surface in a Euclidean space, the osculating plane at a point of (C) Is
normal to the surface. Tkie property, which Is true In the attached Euclidean
space, ie alec true in the Riemanntan epaae.

Osculating Space
105
IV. APPLICATION TO THE THEORY OF SURFACES IN ORDINARY SPACE
99. A surface embedded in ordinary space could be regarded as a 2-
dimensional Riemannian space whose line element 1s the ordinary (Gaussian)
line element of the surface. We have here a aonorete interpretation of the
osculating Euclidean space at a point Η (in this case it 1s the plane).
Let us consider an orthogonal projection of the points of the surface onto
the tangent plane at M. We thus obtain a particular representation of the
surface on a Euclidean space, and the line element of the plane osculates that
of the surface. This becomes evident if the point Μ 1s placed at the origin
of the co-ordinates and if rectangular axes are taken such that the xy plane
1s the tangent plane. Keeping to the classical notations as given by Honge,
2
we have for the de of the surface,
de2 = dx2 + dy2 + (p dx + q dy)2 ,
whilst that of the plane 1s
2 2 2
chc - ax + dyc .
It can be seen that at the origin of the co-ordinates, the coefficients of the
two line elements are equal, as well as their first order partial derivatives.
In a more general way, the point W{x,yta) of the surface could be made
to correspond to the point (x-.*y-.) of the tangent plane obtained by taking
through Η a line with variable direction, but making with the normal to the
surface at Η an angle tending towards zero when M* tends toward H. In
a more precise way, we will take
* - Xy У - У]
"1 — * * ■
l and m being continuous functions adnitting continuous first order partial
derivatives, with the condition that they are zero for χ ■ у = 0. We have
1n effect
x1 - x - Хя , i/1 « у - mz
dx2 + dy2 - (dx - idz - adSL)2 + (dy - mdz - zdm)2
By Ignoring the second order terms in the coefficients of the expansion of the
right hand side, we have
dx\ + dy2 - dx2 + dy2 .
2
It so results that the de of the two surfaces tangent at a point A
osculate at this point if a point correspondence 1s established between the two

106
Oeeutating Spaae
surfaces by the Intermediary of either the perpendiculars to the совпоп tangent
point at A, or the normals projected from points of one surface onto the
other.
100. It was Gauss who first developed the intrinsic theory of surfaces
by studying their properties which only depend on their line elements. To
this extent, he was a precursor of Riemann. In particular, he had Introduced
the concept of geodeeia curvature of a curve traced along a surface. To avoid
all confusion» let us adopt Gauss's approach and denote by —!— the geodesic
curvature of a curve. We reserve the name curvature for the ordinary curvature
of the curve considered as one embedded in Euclidean space, and denote it by
Ρ
The geodesic curvature at a point Η of a curve С 1s, according to the
general theory and No. 99, equal to the ordinary curvature of the projection
(C) of the curve (C) onto the tangent plane at N. If wa now consider the
cylinder projecting orthogonally the curve (C) onto the tangent plane, the
two curves (C) and (C) have the same normal curvature on this cylinder.
As the angle made by the principal normal to the curve (C) with the normal
to the cylinder 1s the complement of the angle V which 1s made by this
principal normal with the normal to the surface, we have the relation
1 . cos'f · V> . ,1, V
pg p β '
The geodeeica of the surface (curves with zero geodesic curvature) are, because
of this, either the lines of the space (- ■ 0), or the lines whose osculating
plane 1s normal to the surface (sin V - 0).
It 1s possible to have another expression for the geodesic curvature that
does not Involve the ambient space. Firstly, let us note that in a
2-dimensional Rlemanrlan space, two vectors χ and x' with infinitesieally near
origins Η and M* are parallel if they make the same angle with the
geodesic MM*. This holds according to whether the direction of the geodesic
stays near enough the same when it 1s displaced on this geodesic. This
established, let us take the Frenet formula
3e Qg 9
where η denotes the unit vector normal to the curve, but tangent to the
surface. The magnitude of the vector Dt 1s equal (in radians) to the angle ε
It 1s what we have called the curvature of the curve, considered as being
traced across the 2 R1emann1an space constituted by the surface.

Osculating Space
107
(angle of contingency) that the tangent at Μ makes with the tangent M*
transported by parallelism from W to M. Consequently, the geodeeio
curvature ie equal to the ratio -3- of the angle of contingency to the arc MM'.
Now to calculate ε, we shall consider (Fig. 7) the two geodesies tangent to
Μ and M' at the given curve, and let Ρ be the point (Inflnlteslmlly near
to Μ and M') where they Intersect. To transport In a parallel sense the
direction of the tangent at M' up to Infinitesimals of the second order,
It Is possible to transfer this direction at first from M* to P. We thus
obtain the direction of the geodesic PM' at P. In the subsequent transfer
from Ρ to Μ', the angle that this direction makes with the geodesic MP
does not vary. Consequently the angle of contingency is none other than the
angle between the intersection at Ρ of the tuo geodesies in question. We
thus obtain the same definition of geodesic curvature as for the Euclidean
plane.
101. We also have a concrete Interpretation of the attached Euclidean
space (here the plane) along a line (C), by orthogonally projecting the
points on the surface onto the developable (Σ) circumscribing the surface
along (C). In effect, the metrics of the two surfaces osculate at each
point of (C) (see No. 99) and on the other hand, the metric of ιΥΊ Is
Euclidean. Consequently, the surface (£1 that unfolds onto a plane,
provides the attached Euclidean space along (C). The result of this Is a more
or less "mechanical" method of transporting by parallelism a vector along a
line (C) on the given surface. If for example, we consider a sphere of
radius R and on this sphere, an arc of a parallel of co-latitude a, then
we could make a transport by parallelism from A to B, the tangent AT at
the meridian PAP* which passes through A (Fig. 8). The developable
circumscribing the sphere Is here a cone, whose generators, limited to the given
parallel of the sphere, have a length R tan a. If β Is the centre at the
centre corresponding to the arc AB, the development will give (Fig. 9) an
arc of a circle of radius R tan α and length R6 sin a. The geodesic
curvature of the parallel Is thus C0R α ; the angle at the centre of the sector
resulting from the development of the cone will be β cos a. The Immediate
result Is that It Is the exact angle made on the sphere by the vector ВТ'
(resulting from transport by parallelism from AT) with the tangent ВТ to
the meridian passing through B. This angle β cos α Is zero If the circle
under consideration Is a great circle of the equator. This pertains to the
equator being a geodesic, the tangent to which Is constantly parallel to It,
the vector AT thus constantly makes a right angle with this geodesic during
Its transport.
Flf. ?·

108
Osculating Space
r.#. ν.
•o*
v*>.
?хУ
It 1s possible to achieve the preceding results analytically by taking
2
the classical de of the sohere with a radius of 1:
2 2 2 2
dsc = d* + sin£e d<r
The calculation of the Chrlstoffel symbols gives
2r2 ■ |Λ| ■ -s1n θ и5 θ
$ - ΙΛΙ
я cot θ ,
The other quantities Г.. are all zero. The equations that express how a
vector 1s transported by parallelism along a parallel are:
df} + X2u>2 - dX1 - X2 sin θ cos θ άφ - 0
dX2 + Χ]ω2 « г£Х2 + X1 cot θ d<b - 0 .
Here we have
9 -α , (Χ])0 - 1 , (X2)0 * 0

Osculating Space
109
Inteqratlng from φ to φ + β, gives at В
X1 - cos($ cos a) , X2 * - j^~— s1n(6 cos a)
which Is very much In accordance with the result obtained geometrically.
102. The method outlined In 99 for obtaining a Euclidean osculating
metric at a point on the surface could be generalised for any n-dlmenslonal
Rlemannlan space. It suffices to Imagine In a Euclidean space of a
sufficiently large dimension N, an n-dlmenslonal manifold having precisely the
given line element. Such a manifold will be obtained If N functions x,,...,
1 η
xN of и ,...,ц can be found satisfying the Identity
dr2 + dx\ + ... + dx\ ■ gtj duO\?
This Identity amounts to a system of %** —i equations with first order
partial derivatives of N unknown functions. This system will certainly be
compatible (In all events a deeper Investigation shows It to be so) If there
are no more equations than unknowns, In particular If N Is taken to be equal
to nlaplS
*
See M. Janet, Annalee Soa. Pol. Math 5, 1926, pp. 38-43, and E. Carta η In the
same journal, No. 6, 1927, pp. 1-7.

Chapter V
GEODESIC SURFACES
THE AXIOM OF THE PLANE AND THE AXIOM OF FREE MOBILITY
I. GEODESIC SURFACES AT A POINT; SEVERI'S THEOREM
103. If through a given point A In Rlemannlan space, we take the
various geodesies* tangent to this point, with the same given plane element, we
obtain a surface that Is said to be geodesic at A. To see that a given
surface Is geodesic at one of Its points. It therefore suffices to consider the
geodesies tangent to the surface at that point; they must be completely
continuous there.
A geodesic surface at A has at this point Its two principal zero
curvatures, since the normal curvature Is zero when a passage Is made from the
point A to a point A' Inflnlteslmally near on the surface. If we then
transport by parallelism from A to A* a vector tangent to the surface at
A, It remains tangent to the surface at A' (to an Infinitesimal of order
greater than the second). It Is possible to derive from this a geometrical
construction due to F. Severl** concerning the parallel transport of a vector.
To transport by equipollence a vector χ originating from A to a point
A* infinitesimal ly neart is to construct the joining geodeeic AA' ae well ae
the geodeeic surface at A tangent to λ with this geodesic and to the vector
x. The required vector x' is tangent at A* to this surface, and makes with
the geodesic AA\ produced on the other side of A1, the same angle that the
given vector χ makes with the geodesic AA'.
The last part of the above statement results from the direction of the
geodesic AA* at A', being parallel to its direction at A and that In the
parallel transport, the angle between the two directions Is preserved.
It Is Important to note that If we were to transport by parallelism the
vector χ along a finite arc of the geodesic AA* (sufficiently produced),
the vector obtained would no longer be, In general, tangent to the considered
geodesic surface, since there Is no reason, a priori* to suppose that this
surface, geodesic at A, Is geodesic to the other points of the geodesic AA*.
If two surfaces. Intersecting along a line (C), are geodesic at all points
This, and In what follows. Involves the geodesies of the space or straight
lines.
**
F. Severl, Sulla curvature delle euperfioie e varieta (Rend. Clrc. matem
Palermo, 42, 1917, pp. 227-259).
Ill

112
Geodesic Surfaaes
of (C), then they Intersect at a constant angle, as the line С could be
regarded as a line of curvature of each of the surfaces (see No. 96).
II. TOTALLY GEODESIC SURFACES: PLANES
104. A surface which Is geodesic at each of Its points Is said to be
totally geodeeia. It then enjoys the characteristic property that every
geodesic which 1s tangent to 1t 1s completely contained within It.
An equally characteristic property Is that every geodesic that has two
of Its points (sufficiently near) on the surface Is also completely contained.
Let A and A' be these two points. From the point A there stems an
Infinity of geodesies along the surface; these cover all of the surface within a
sufficiently small neighbourhood. One of them passes through A', and as
there passes only one geodesic through two sufficiently close points In the
space, it is precisely the geodesic in question. Similarly, the converse can
be proved.
Totally geodesic surfaces therefore possess the characteristic properties
of the plane In Euclidean space. We would reserve the name of planee for then.
But we shall see that tha eartetenae of pianos in a Ritmannian epaae ie взвалр-
tional.
105. A totally geodesic surface has the three following, equivalent
properties:
Firstly, it has zero principal curvature at each of its points.
Secondly, the unit vector normal to the surface remains normal if a transport
by parallelism is made in any manner whatsoever along the surface.
Thirdly, every vector tangent to the surface remains a tangent if similarly
transported as in 2.
The second property Is an Immediate consequence of the first which In
turn can be derived from the second. As for the second and third, these are
manifestly equivalent to each other. We are going to show that the third
property Is characteristic of the totally geodesic surfaces. This results In each
of the first two being equlvalently characteristic of these surfaces.
Let us assume that each vector tangent to a given surface S renalns so
If It Is transported by parallelism, Its origin describing an arbitrary curve
of the surface. The result is that the acceleration of a point describing
such a curve of the surface Is always tangent to the surface. If the
coordinates of a point on the surface are expressed In terns of a function of
This notion Is due to J. Hadamard {Bull. Soo. Math. 2nd series 25, 1901,
pp. 37-40).

Geodeeie Surfaoee
113
two parameters a, 0, It will suffice to show that the acceleration of the
moving point is zero In terms of two second order differential equations in
α and 0. There exiate therefore ση the eurfaoe a gecxleeic pausing through
an arbitrary point of thie eurfaoe and having at thie point an arbitrary
direction tangent to the eurfaoe; consequently the eurfaoe ie totally geodeeie.
Verifying this by calculus Is easy. Let us assume, without loss of
generality, that the surface Is defined by the equation w ■ 0. In determining
the spatial geodesies traced across the surface, the following three equations
must be satisfied:
сиг %3
Γ3
du dJ
da da
du1 dJ
du da
A1" dJ
da da
Because the surface Is totally geodesic. It Is then necessary and sufficient
to have
Γ13Γ Γ132- Γ22" ° ■ (D
These equations Indicate that the prlnlcpal curvatures are zero, whether the
absolute differential of each of the vectors ву &г tangent to the surface,
Is a vector tangent to the surface
De1 ■ (Г^Л1 + т{гаиг)еу + (ifjA1 + Ι^ΛΖ)·Ζ .
or, the absolute differential of the vector normal to the covarlant components
X, ■ X- ■ 0, X- ■ 1, is normal to the surface
DX] ■ dX} - XiuiJ ■ -ΐή ■ - ifjdb1 - ^A2 B 0
DX2 ■ dX2 - Χ£ω2 ■ -ω3 ■ - Ij^du1 - Г*гаи2 ■ 0 .
106. Another Important property of totally geodesic surfaced Is the
following: If we develop in Euclidean epaoe any one of the curoee traced
on a totally geodeeie eurfaoe, a plane curve ie obtained. In effect, with
the absolute differential of the unit tangent vector being tangent to the
surface, then the principal normal Is tangent to the surface. The unit vector
taken on the binormal Is then normal to the surface and Its absolute

114
Geodeeia Surfaoee
differential Is zero; the torsion of the curve is thus zero. In the subsequent
development, the curve, having zero torsion, becomes planar.
Moreover, the oonveree ia true. Let us assume that for evtry curve traced
across a surface, the torsion is zero, or what amounts to the sane thing, that
the velocity, the first and second order accelerations of a moving point, taken
In any manner whatsoever on the surface, are constantly coplanar. Let the
surface be defined by u -0. It Is always possible to imagine a second motion
across the surface such that at any given Instant, the quantities ~j|- and
2 г
3-^2 have the same numerical values as for the first motion, whereas the quan-
dt <?λ d*2
titles =-£* and =-*Ц- vary from the first motion to the second by arbl-
dt dt
1 2
trary quantities α and α respectively. The result of this Is that the
1 2
arbitrary vector (a ,oc) tangent to the surface 1s coplanar with the
velocity and acceleration of the first motion. Consequently, this acceleration
Is In the tangent plane to the surface and all the lines traced across the
surface are asymptotic; the surface Is thus totally geodesic.
107. From this, we are going to deduce an Important theorem due to
G. Rlcd.*
Theorem. If, in Rienznnian epace, there exiete a one parameter· fcrtily of
planee, their orthogonal trajeatoriee eetablteh an ieametrio point oorreepon-
denoe between the different planee of the femily*
In effect, let (P) be a plane In the family, and let (P1) be a plane
Inflnlteslmally near. Let (C) be any curve traced In the plane (P), and
let (C) be the locus on (P1) of the orthogonal trajectories taken through
the different points of (C). Let us make the representation In the adjoining
Euclidean space along (C). In this representation, the curve (C) Is planar
and the curve (C) Is produced by taking through each point of (C), an
infinitesimal length normal to the plane of (C). Consequently, the first
variation of any arc length of (C) is zero when going from (C) to (C).
On the other hand, the length of (C) Is preserved In the reoresentatlon
up to Infinitesimals of the second order. It can be seen that In Riemannian
space, the first variation of the arc length of (C) Is zero when going from
the plane (P) to the neighbouring plane (P1). This 1s what had to be
proved.
From this, something Interesting comes about. For one of the planes
(P ) of the family, let us take any co-ordinate system, u, u, and denote
by d a variable parameter Individualising the different planes of the
G. Rlcd, Fomole fondamentali nella teoria generate aelle varieta β delta
loro ourvatura (Rend. Ace. Lincei 12, 1903, pp. 409-420.

Geodeeio Surfaoee
115
family. The spatial line element 1s then of the form
da2 - da2 + H(u,u,u)dJZ , (г)
2
on denoting by do the line element of the plane Ρ ;
do2 ■ i{utv)du2 + ZF{utv)dudv + G(u,u)duZ . (3)
We could, however, derive this result by calculus if we take the orthogonal
trajectories of the planes of the family as the third family of lines with
co-ordinates (u ■ cost., и - const.), and the planes themselves as the third
family of surfaces with co-ordinates (u ■ const.) The spatial line element
would satisfy the conditions
*i3 " *гз " ° · (4>
Because the surface и * const. 1s totally geodesic, it is necessary and
sufficient to have
Г? ■ Г* ■ Г* ■ 0 ,
r r гс
or, on account of (4)
ri3i " Г1зг-Ггзг" ° ·
or, using the Chrlstoffel notations,
[11 3] ■ [12 3] ■ [22 3] ■ 0 .
These equations reduce to
^11 . ^12 n ^22 , n
эи au au *
and the da form Indicated above may be derived.
The line element so obtained only contains one arbitrary function of
three arguments (and functions of two arguments). The result 1s that Rieman-
nian spaces which admit families with a planar parameter are exceptional,
2
since Riemann postulated that the most general da in three variables
depends on three arbitrary functions of three variables (there are six
arbitrary coefficients, but the possibility of any change in co-ordinates reduces
the number of these functions to three).

116
Geodesic Surfaces
III. THE AXIOM OF THE PLANE AND THE AXIOM OF FREE MOBILITY IN SPACE
108. In ordinary space, a plane always passes through any three given
points, or what amounts to the same thing, there always passes a plane through
two Intersecting lines. We shall say that a R1enann1an space satisfies the
Axiom of the Plane if it has the following property:
Through every point of the враге and tangentially to every plane element having
thie point as an origin, there passes a totally geodeeio surface, that is to
say, every geodeeio having tut? of its points on that surface, is completely
contained in it.
Later we shall determine every R1emann1an space satisfying the Axiom of the
Plane. In the meantime, we shall show that they have another very Important
property In common with Euclidean space.
Euclidean space admits an Infinity of Isometric point transformations;
in other words, it 1s always possible to displace a figure without it changing
in any way, so as to make any point A on the figure coincide with any given
point B, and any two half lines stewing frora A with two such lines from
B, to make the same angle between each other as in the case of the first two.
It 1s on the existence of these displacements, or what amounts to the
same thing, on the Idea of equality, that a large part of elementary geometry
1s founded.
We shall say that a R1emann1an space satisfies the Axiom of Free Mobility
if it admits displacements that are non-deforming, and have the same degree of
generality as those of Euclidean space.
We are going to show that in Riemannian space, the Axiom of the Plane
implies the Axiom of Free Mobility, and vice-versa.
109. First of all, we are going to show that If all geodesic surfaces
at a particular point are totally geodesic, the space has free rrvbility about
A, that is to say, it always admits an isometric point transformation leaving
the point A Invariant, and transforming any two directions sterming from A
Into two other directions with the same angle between them as for the two
given directions.
Clearly, If such an Isometric point transformation exists, it 1s well-
determined, for in the Euclidean tangent space at A, it necessarily appears
as a rotation about an axis through A. As the transformed direction from
another 1s well defined, to every point Μ situated on the geodesic sterming
from A in the first direction, there corresponds a point H' situated on
the geodesic stemming from A in the transformed direction and at the same
distance from A as the point H. This assumes that the Isometric
transformation 1s direot, that is, it preserves the sense of the trlhedrals attached to
A.

Geodesic Surfaces
117
Having established these preliminaries, let us assume that all geodesic
surfaces at A are totally geodesic. We shall consider a sphere centred at
A, that Is, the locus of points obtained by taking from A, on the different
geodesies sterming from A, a constant length R. Let us call a great circle
of this sphere the section of the sphere through a plane passing through A,
and consider on the sphere two great-circular arcs MN, H'N' corresponding
respectively, to two equal angles HAN, Η'AN' at the centre.
It 1s possible to trace across the sphere, an arc KM1 normal at Η to
the great circle Ю, and Η' to the great circle HH1. We shall consider
the family of planes obtained by taking on the curve KM1 an arbitrary point
P, and constructing the geodesic surface containing the geodesic AP and
which 1s normal to the curve W at P; this 1s possible since the radius
AP 1s normal to the sphere at Ρ (see No. 95).
The orthogonal trajectories of the planes of the family define an
Isometric point correspondence on these planes. Now at the point Η of the plane
ANN (which belongs to the family) there corresponds the point H* of the
plane AM'N1 (which also belongs to the family). To the geodesic AN, there
corresponds the geodesic AN1 which makes the same angle with AN1 as AN
does with AH, and consequently the point N1 corresponds to N. The
Isometric correspondence thus Induces a passage from the arc of the great circle
MN to the arc of the great circle H'N*.
From this, it results that on every sphere centred at A, eqial angles
at the centre correspond to equal arcs of great circles (i.e. they are equal
in length).
Now let us consider the point transformation of the Rlemannlan space
which results, as was explained, from a rotation about A in the Euclidean
tangent space at A. Let Η and N be two neighbouring points situated on
the same geodesic steeming from A, and let H1 and N1 be the transformed
points; they are also on the sane geodesic, and 1t can be seen that H'N* ■ HN.
Now let Η and N be two neighbouring points situated on the sane sphere
centred at A and let H1 and N1 be their transformed points also on the
same sphere; again MN ■ H'N1, since the corresponding angles are evidently
equal. Since every elementary displacement in Rlemannlan space can be regarded
as the resultant of two elementary orthogonal displacements, one being radial,
the other being normal to the radius vector stemming from A; this proves that
the point transformation in question is clearly isometric.
We might add that the above reasoning would apply without modification
to the point transformation that results from a symmetry in the tangent space
at A.
If therefore the surfaces at A are totally goedesiot the space admits
3 3
• direct point transformations and · inveree point transformations, all
of which are isometric and leave the point A fixed.

118
Geodesic Surface в
110. Let us now prove the converse. We allow the space free Mobility
about A. It results Immediately that, on every sphere centred at A, two
elementary arcs corresponding to equal angles at the centre are equal.
Consequently, let us consider a plane element stemming from A and the symnetry
with respect to this plane element in the Euclidean tangent space at A. It
Involves a point transformation of the Rlemarmlan space, whereby a point Η
1s taken to the point H1, such that the two geodesies AM and AH1 have,
at A, their directions symmetric with respect to the given plane element,
and such that the two lengths AH and AH1 are equal. This point
transformation, preserving elementary arc lengths traced across a sphere centred at A
as well the lengths of elementary arcs normal to this sphere, 1s Isometric.
In other words, the existence of « direct transformations about A Implies
the existence of as many inverse Isometric transformations.
111. The converse that we are about to prove results in the fact that
if a Riemannlan space satisfies the axiom of free mobility, it satisfies the
axiom of the plane, since all the geodesic surfaces at any point of the space
are totally geodesic. Conversely, if a Riemannlan space satisfies the axiom
of the plane, then it also satisfies the axiom of free mobility. Let A and
A1 be any two points (sufficiently near), along with two equal angles
B'A'C; α rotation (Isometric} conveniently chosen about the centre of the
geodesic AA1, will take A to A1, and a second rotation about A* will
lead to a coincidence of the two given angles.
#
112. We are now going to prove an Important theorrm due to F. Schur,
namely if too certain pointe, A and В {sufficiently near) exist in the
Riemannian epaoe, euah that every geodeeio surface at one of thee· pointe ie
totally geodeeio, the space will eatiefy the axiom of the plane. Moreover,
it ie possible to repreeent thie in ordinary epaae by saying that every
spatial geode&ia is represented by a straight line.
The first part of this theorem could be re-stated as follows:
If a Riemannian space enjoys free mobility about two certain pointe,
than it satisfies the axiom of free mobility.
From this point of view the theorem would not be difficult to prove. If
1n effect, Η 1s any point (sufficiently near to A and B), then the space
admits a rotation through any angle about the geodesies HA and about NB.
Now in the Euclidean tangent space at H, every rotation about A could be
obtained by the composition, 1n a convenient order, of a sufficient number of
F. Schur, Ueber den Zneammenhang der Fame oonstanten Krwmungemaeeee mit
den projeotiven Ramen (Hath. Ann. 27, 1886, pp. 537-567).

Guodaeio Suvfaoes
119
rotations about the two given straight lines steaming from H. These
compositions, made In the Rlemannlan space Itself, give rise to Isometric
transformations which prove the free mobility of the space about M.
113. We are going to prove Schur's theorem from a purely projective
point of view. Every geodesic stemming from A could be analytically defined
by Its direction parameters яг', y\ «', at B. Therefore at every point Η
of the space (sufficiently near to A and B) there are attached six numbers
Xt y* *; χ1· y', *'» of which the first three, as well as the last three,
only occur by their mutual ratios. This In practice results In four
co-ordinates, between which there necessarily exists one relation.
To obtain this relation, let us note first of all, that every plane
(totally geodesic surface) passing through A Is represented by a linear,
homogeneous equation In x, у and я. Similarly, every plane passing through
В Is represented by a linear, homogeneous equation In х1, у' t and *'.
Consequently every plane containing the geodesic AB will be represented In
two different ways: on one hand, by an equation of the form
α-,χ + b-,y + a-.* + m[a^e + ЬуУ + Оуя) ■ 0 , (5)
where π Is a variable parameter with the plane In question; on the other
hand, by an equation of the form
a\x* + by + oj*' + m'iaj*' + b^' + <··«') ■ 0 . (6)
where m" is an equivalent variable parameter with the plane in question.
The equations
α-,χ + b,y + ο-,λ в 0 , a^c + ЬуУ + ОуЯ * О
define two distinct planes [basic ρ lane в) of the sheaf formed by the planes
containing AB. The same applies to the equations
a\x* *b\y* + o\a* ■ 0 , α£χ' + Ь^' + о^в' - 0 .
There is nothing to prevent us from assuming that the same two basic planes
were chosen at A and B. At the same time It can be assumed that a third
defined plane corresponds to the value 1 for each of the parameters m and
m\
With this established, the quantity
a\xl + b\yl + oj«'

120
Geodesic Surfaces
takes the same value for all points which correspond to the same numerical
value of
α-,χ + b-,y + о-, в
a~c + ЬуУ + Oy* *
therefore the two quantities are functions of each other; this In turn.
Involves the sought-after relationship between the co-ordinates attached to
an arbitrary point. Now It Is easy to see that the στσββ ratio of four planee
paseing through AB ie the вате at A and at B. Let us consider a surface
(S) cutting the geodesic AB at a point С Every point of this surface
could be defined by the homogeneous co-ordinates x, yr ж of the geodesic
which joins It to the point A. If (s) Is cut by four fixed planes passing
through AB, then we shall have four curves stenming fron C, which are
defined by the equation
α-,χ + b-,y + σ-,Λ + тЛа~с + Ь~У + Оря) * 0
Consequently the оговв-ratio of theee four curves at Ζ ie equal to the
ai-voa-rativ of the four valueo of m. Effectively, if it is as suae d that at
С, «/О, then it may be assumed that there, «- 1 and by putting - β Χ,
И- * Υ, the directions of the four curves at С are defined by the equation
a^d* + bydi + m_.{a2dX + bydi) ■ 0 .
Similar reasoning shows that the cross ratio is equal to the cross ratio or
the four values π1, of π1 which correspond to the four planes. The
relation existing between the parameters m and m' of a similar plane
containing AB ie therefore homographio.
As there corresponds to the values т-0,«\1 the same values n1 = 0,»,1,
the nomographic relationship reduces to π1 - т. In other words, we have
between the six quantities χ» ι/, я, х\у\я\ the relation
α-,χ + byy + σ-,Β ajje' + b\yx + <?j«'
a^x + byy + ел aW + b^y' + olz*
Me then have
α-,χ + b.y + o,« 3 p{alxl + b\y* + o|«') ,
a~x + byy + Ojjs. B p(aix' + Ьхуух + о£я')
Let us now take a linear form a-x + Ьл + о-я, linearly Independent In the
two forms

Geodeeia Surfaces
121
α,χ + b*y + a*z and a^c + b-y + а^я
Let us Introduce a similar form air' + b^y* + ^ϊ*'· linearly Independent
In the two forms
ajx' + b\yx + а\л* and a^' + ЬЛу' + oLb*
Finally we have
X * a,x + b*y + ο-,Β * p(ajx' + b\yl + ΰϊβ')
Υ - a^e + Ь^ + u^ ■ p(alx' + b£y' + o^1)
(8)
Ζ ■ α-χ + Ь~у + ^т8
J - δ(α^' + Ь^' + a'js1) .
Every point of the space Is completely determined by the four quantities
Χ, Υ, Ζ, Τ (or rather by their ratios) since knowing Χ, Υ and Ζ gives
x, у, я, and knowing Χ, Υ and Τ gives, up to a factor, х', у*, a1.
114. With the co-ordinate system thus obtained, every plane равНлд
through A ie defined by a linear equation in Χ, Υ, Ζ and every plane
through В by a linear equation in Χ, Υ, Τ. Let us now take an arbitrary
spatial geodesic; this geodesic, together with the point A, determines a
totally geodesic surface, which Is defined by a linear equation In X, Y,
Z. With the point B, the geodesic determines. In the same way, a totally
geodesic surface defined by a linear equation In Χ, Υ and T. Thus every geo-
deeio ie defined by two equations of the first degree in Χ, Υ, Τ.
The converse is true, since a system of two equations of the first order
Is equivalent to a system formed by an equation in Χ, Υ, Ζ and an equation
In Χ, Υ, Τ. They therefore define a curve of Intersection of two totally
geodesic surfaces, which Is necessarily the geodesic joining two points of
this curve.
If we regard Χ, Υ, Ζ, Τ as the homogeneous co-ordinates of a point of
the ordinary space, we see then that the region of the Riemannian epaoe
neighbouring the points A and В admits a representation on the ordinary epaoe
in which the geodesies are represented by etraight linee (geodmeiaue repreeen-
tation).
The axiom of the plane Is Immediately deduced from the preceding result.
The surface admitting any plane of the ordinary space for representation Is
In fact a totally geodesic surface, since from each of Its points there stems
an Infinity of geodesies situated on the surface and these geodesies are
tangent at this point to a similar plane element (see Note I).

122
Geodesic Surfaces
We may see In particular that the axiom of the plane leans to the possi~
bility of a geodesic representation of the Riemannian epaoe on the ordinary
space; the oonveree is true, as we are going to see.
Therefore, if we consider the following three properties of a Riemannian
space:
I. It satisfies the axiom of free mobility.
II. It satisfies the axiom of the plane.
III. It aamite a geodesic representation on ordinary space,
then each of these properties Implies the other two.
115. The above theorems extend to the case of any number of dimensions
η > 3. Let us restrict our attention to the general definition of geodesic and
totally geodesic manifolds, and to prove a theorem relating to these manifolds.
A p-dlmenslonal manifold V passing through a point A Is said to be
geodesic at A If It contains all of the geodesies stemming from A and
tangent at this point to a similar p-dlmenslonal plane element Ε . A manifold
V will be said to be totally geodesic If It Is geodesic at each of Its points,
or equlvalently If every geodesic having two of Its points there Is completely
contained within It.
Let us assume that all geodesic surfaces (two-dimensional manifolds) at
a point A are totally geodesic. He are going to show that the geodesic
men 1 folds V (p>2) at A are also totally geodesic. Effectively, letting
К and N be any two points of one of these manifolds; the geodesies AH and
AN determine at A a plane element E0 contained In the plane element Ε
2 Ρ
which defines the manifold V . There exists a surface S geodesic at A,
tangent to E?, and this surface contains the points К and N; being
totally geodesic, It contains the geodesic MN entirely. The manifold V
containing the surface S thus also contains the geodesic MN; It Is
therefore totally geodesic.
Conversely, let us assume that all geodesic manifolds V (p>2) at A
are totally geodesic. We consider a surface S geodesic at A and tangent
at A to the plane element E?. This plane element could be regarded as
the Intersection of an Infinity of p-dlmenslonal plane elements Ε ,
consequently S could be regarded as the Intersection of an Infinity of geodesic
manifolds at A. Therefore let Μ and N be two points belonging to S;
they belong to each of the preceding manifolds V and thus the geodesic MN
belongs entirely to each of them, It being assumed by hypothesis to be totally
geodesic. It therefore belongs entirely to their comon Intersection, that Is
to say, the surface S, which Is Itself therefore, totally geodesic.
116. In the case η-2, of the three properties mentioned In No. 114,
the second Is Invalid. The equivalence of properties I and III is again true,
but It depends on reasoning far more technical than In the и ■ 3 case to

Geodesic Surfaces
123
prove geometrically. Moreover» In Weyl's theory of spaces which generalise
those of Rlemann, the equivalence of I. II, and III Is again exact for n-3,
but for η ■ 2, the equivalence of I and III no longer holds.
In seeking to produce a geodesic representation of a given surface on a
plane, Beltrami discovered the Impossibility of such a representation for an
arbitrary surface. The only surfaces for which the representation Is possible
are those with total constant curvature, Isometric to a sphere (with real
or purely Imaginary radius); they are also the only ones that satisfy the axiom
of free mobility.
E. Beltrami, Rieolunione del problema riporture I punti di una euperfioie
eopra un piano in modo aha le linee geodetiche veggano rappreeantate вы lime
vette* Ann di Matem 1st serle, 7, 1865, pp. 185-204; Opere matem I, Kllan,
1902, pp. 262-280.

Chapter VI
NONEUCLIDEAN GEOMETRIES. SPHERICAL SPACE.
ELLIPTIC SPACE. HYPERBOLIC SPACE.*
117. In this chapter we are going to review a class of Rlemannlan spaces
enjoying the property of atolttlng a geodesic representation In ordinary space;
these spaces are said to have constant curvature. These spaces are
furthermore said to be non-Euclidaan; the geometry of these spaces Is said to be
non-Euolidean geometry. We shall commence with the simple case of n-2.
I. TWO DIMENSIONAL SPHERICAL GEOMETRY
118. Let us consider a sphere of radius R In ordinary space. Every
point of the sphere could be defined analytically by Its co-ordinates x. у, ж
with respect to three rectangular axes» having for Its origin the centre of
the sphere. These quantities are subjected to the relation
x1 +j,2 + «2« R2 ; (1)
we have then
ds1 * dx2 + ay2 + dzZ (2)
The lines which here play the same role as straight lines (geodesies)
are the great circles of the sphere. It Is evident that the sphere atolts a
geodesic representation on a plane. It suffices to make a central projection
on a plane (the point of view being the centre of the sphere). It Is also
evident that the axiom of free mobility Is verified, since we could always take,
by rotation, any point of the sphere to any other point In such a way that any
direction stemming from the first point coincides with any direction stemming
from the second point. All the axioms on equality, stated In the foundations
of Euclidean geometry, are verified In spherical geometry.
The fundamental difference between the geometry of the Euclidean plane
and the spherical geometry Is the following: It Is possible to pass a
"straight" line (great circle) through two points of a sphere and then through
The literature relating to the non-Euclidean geometries Is quite vast. We
might consult In particular the fine papers of F. Klein, entitled: Uber die
eogenarmte nioht-euolidieohe Geometrie (Math Ann t. 4, 1871 and t.6 1B73).
See also P. Barbarln, Le Geotmtrie поп eualidienmt following notes by A. Buhl
(collection Sciential Paris, Gauthler-VIUars, 192B).
125

126
Νση-Euolidean Geometries
an Infinity of then. This occasion presents Itself when the two points are
diametrically opposed.
There exists some other essential differences; for example, the two
dimensional space constructed by the surface of the sphere Is finite (and with area
4ttR ). The "straight lines" (great circles) are closed curves of finite
length 2nR.
These differences are,nevertheless, more superficial than essential,
for we have seen In Chapter III that certain looally Euclidean epaoeo exist
along with Euclidean space.
A local difference between the sphere and the Euclidean plane Is given
by the sue of the angles of a triangle. On the sphere It exceeds π, and
the difference between this sum and π Is equal to the quotient of the sur-
2
face of the triangle by R .
Let us point out again that a circumference has two centres and
consequently two radii r and nR - r, that the radius of curvature of such a clr-
y wR
conference Is R tan =- , and lastly, that a circumference of radius V has
a zero curvature; In other words, It Is a straight line. All the radii are
then perpendicular to the straight line.
II. TWO DIMENSIONAL ELLIPTIC GEOMETRY
119. Let us construct the geodesic representation of the sphere on a
plane (P) by a projection made from the centre 0 of the sphere. Every
point of the sphere gives one and only one point of the plane (at a finite or
Infinite distance), but conversely a point of the plane originates In two
diametrically opposed points of the sphere. Let us define tn the plane, the {non-
Euolidean) elementary distance between two Inflnlteslmelly close points К
and N by the ordinary distance (taken on the sphere) of the two
corresponding points H' and N' of the sphere; this distance does not change If the
two points K' and N' are replaced by the two diametrically opposed points
which provide the same points Μ and N In the plane (P).
The two dimensional Rlemannlan space so defined ts called the elliptic
plane and the geometry of this plane Is elliptic geometry.
It Is essential to note that the elliptic plane Is a closed manifold;
the points which In the ordinary sense of the word are at Infinity are
ordinary points which are, from the point of view of elliptic geometry, at a
finite distance (and form a straight line which corresponds to the great circle
of the sphere parallel to the plane P). From the topological point of view,
the elliptic plane Is thus Identical to the projective plane.
The topological differences that exist between the sphere and the elliptic
plane could be revealed by noting that It Is possible to correspond to every

Νση-Euolidean Geometriee
127
point of the sphere one and only one [straight) half-line stemming from 0,
whereas to every point of the elliptic plane there corresponds one and only
one straight line passing through 0. Let us consider then a plane (π)
passing through 0. The manifold of the half lines from 0 Is divided by
this plane Into two separated regions; It Is not possible to pass continuously
from one half line situated on a certain side of the plane without crossing
the plane. In contrast, the plane (π) does not separate the manifold of
the straight lines passing through 0 Into two distinct parts; we could pass
continuously from any straight line to another passing through 0 without
ever crossing the plane (π). In other words, a "straight" line (great circle)
of the sphere divides the sphere Into two distinct parts, whereas that of the
elliptic plane does not divide the plane Into two distinct parts. However we
can see this In the case where the straight line Is the straight line of
ordinary Infinity. We can, In fact, pass from a point of the plane, to any other
point without crossing the straight line at Infinity.
120. If the elliptic plane Is locally Identical to the sphere, there
nevertheless exists, as we are about to see, essential differences between the
two manifolds. Here are some others:
Through any two points of the elliptic plane there always passes one and
only one straight line and conversely any two straight lines always meet at one
and only one point. The firet axiome of Eualid&an plane geometry are therefore
true for the elliptic planet We might ask when elliptic geometry differs
from Euclidean geometry In the collection of theorems that are usually
proved In the texts of geometry. This difference clearly appears at the
moment when, wiavoldedly, the Euclidean postulate Is Introduced. We In fact
prove Immediately before hand that through every point taken outside a straight
line, there Indeed passes a parallel to that straight line. Now this theorem
Is false In elliptic geometry as two straight lines always have a coewon point.
The proof of the theorem In question depends on a preceding theorem according
to which one cannot drop a perpendicular froma given point ontoa straight line.
Let us recall the proof of the second theorem. Let 0 be a given straight
line and A an extreme point (Fig. 10). Join A to an arbitrary point Μ of
the straight line 0 and construct at Μ on the other side of the straight
Hne D an angle DMA1 equal to the angle DMA.
Finally, let us take MA' - MA. If N Is any other point of the straight
line 0, we see Immediately that the triangles AMN and A'MN are equal,
having an equal angle between two equal sides In turn; consequently, the angles
ЙнА', ΜΚλ are also equal. This established, If the straight line AN Is
perpendicular to the straight line 0, the two angles MNA and MNA' are
supplementary and the points A, NA' lie on a straight line; the converse Is
true. Hence there exists only one perpendicular dropped from A onto the
straight line 0, I.e. the line AA'.

128
Non-Euclidean Geometries
The preceding conclusion Is tenable only when the point A' differs from
the point A. Now this last situation can not arise In the elliptic plane
which ie not divided into two distinct regions by a straight line. If A' Is
Identified with A, all straight lines passing through A are perpendicular
to the straight line 0. This is In fact what happens when we take In the
elliptic plane a point and an elliptic straight line arising fro* a point of
the sphere and its polar great circle. In the elliptic plane every straight
line can be regarded as a circumference of radius -=- of which the centre Is
the pole of the elliptic straight line. In passing through the Intermediary
*p
of the sphere, one easily sees that the circumference of radius г < -V has
length 2nR sin (j£). When r tends to ~ , this length tends towards 2ttR,
but the length of the maximum straight line only amounts to nR. For the
area bounded by a circle of radius r Is 4nR sin (4jr) ; It tends to 2ir(r,
the total area of the elliptic plane, when r tends to V .
121. If we take the plane (P) tangent to the sphere at a point A,
the projection being made from the centre 0 of the sphere (situated on the
normal to the sphere at A) carries [In the plane (P) assumed to have been
given an arbitrary Euclidean metric] a Euclidean metric osculatory to the metric
of the sphere (see No. 99) and consequently to that of the elliptic plane. We
then have in this plane (P) a representation of elliptic geometry with
preservation of straight lines and a metric osculating the given metric at the
point A (with consequent presentation of the curvature of the curves passing
through A). We can actually see something else. A small circle of centre A
Is represented by a circumference of centre A; now the radius of geodesic
curvature of the small circle B'C of the sphere (Fig. 11) Is equal to the
radius AB = ВС of the circumference ВС. Consequently, the representation
in question preserves the curvature of circumferences of centre A. We can
see quite clearly how the radius of curvature Increases from zero to +™ when
the (non-Euclidean) radius Increases from zero to ttR/2.

Ron-Euclidean Geometries
Kff. "
122. We can propose to define directly the (non-Euclidean) distance
1 *
between two points In the elliptic plane of curvature -y by means of purely
projective concepts. To this extent, we could define analytically a point of
the plane by the rectangular co-ordinates x, y% χ of one of the corresponding
points of the sphere (normal co-ordinates). They are, In short, In the plane
of projective co-ordinates, but subjected to satisfy the relation
x2+*242-R2 ·
With this established, let us consider two points H,H' of the plane with
co-ordinates x, у, χ and x1, y', x* respectively. We evidently have, on
denoting by d their non-Euclidean distance,
R2 cos | β xx' + yyl + xxl
The point In question Is to Interpret geometrically the right hand side of
this relation.
The Isotropic cone having for Its vertex the centre of the sphere,
describes in the plane (P) a conic Γ (said to be the absolute) with
equation
2 2 2
Let N, and N. be two points (Imaginary conjugates) where the straight line
MM' meets the absolute. The co-ordinates of every point of W could be
taken under the guise
*
We recall. In a menner of speaking, the total curvature of the sphere from
which the elliptic plane has been derived.

130
Hon·Euclidean Geometries
χ + λχ' , tf + λ^', л + λζ'
with a variable parameter λ. At the point M, there corresponds the value
zero, and at the point H', the value » of the parameter. Let λ, and λ.
be the values corresponding to the points N, and N.. They are given by
the second order equation
λ2(χ*2 + уЛ + χ'2) + 2λ(χχ' +УУ1 +ав') + χ2 + у2 + я2 ■ 0
or alternatively
λ2 + 2λ cos I + 1 ■ 0
The cross ratio (ΗΗ'Ν,Ν.) Is equal, as we know, to the cross ratio of the
*1
four numbers 0, «, λ-j, λ-. I.e. to y- , a calculation easily yields
^ - e2£ ί
Consequently the distance d could be defined by the formula, due to Cayley,
d = £ log (ΜΗ*ΝΊΝ2) . (3)
which Involves the Naperlan logarithm of the cross ratio of the four points
H, M\ Hy N2. The length d thus defined Is the Cayleylan distance of the
two points with respect to the absolute Г.
We could derive from this formula another Immediate consequence. If
through a point 0 (that could be assumed to be the centre of the sphere)
we take any two straight lines, then the angle between these two lines Is equal
to the quotient by 2t of the cross ratio of the two given lines and of the
two Isotropic straight lines stemming from 0 and situated In the plane of the
two given lines. This theorem is due to Laguerre end provides for ordinary
geometry a projeotive definition of the angle.
Let us return to the elliptic plane. We can see that two harmonic conju-
ttR
gate points with respect to the absolute are at a distance -j- , one from the
other. The cross ratio (MH'N.NJ Is then In fact equal to -1 whose logarithm
Is tTi. The pole of a straight line Is therefore Its pole (In the usual sense)
with respect to the absolute.
In the same way, we could projectlvely define. In the elliptic plane, the
angle between two straight lines stemming from a point A. This angle,
depending only on the numerical values at A of the coefficients of the line element
of the elliptic plane, could be defined as the quotient by 2t of the logarithm
of the cross-ratio of the two qlven straight lines end of the two Isotropic
straight lines stemming from A. Mow a straight line AA' Is Isotropic If

Νση-Eualidean Geometries
131
the two points of Intersection Β,Β- with the absolute Γ are Identified
[for then the cross-ratio (ΑΑ'β,β.) will be equal to 1 and the Cayleylan
distance AA* will be zero]. Consequently, the angle between two straight lines
stemming from A is the quotient by 2t of the logarithm of the arose-ratio
formed by these two lines and the two tangents taken from A to the absolute.
We shall note that if the curvature -y of the elliptia plane is equal
to 1, there is a perfect duality between the idea of distance and that of
angle; the distance between two points transforms by duality In the angle
between the two straight lines.
123. We can set out to find the equation of a circle. Let a, b, a be
the normal co-ordinates of a point A and x, у, χ the normal co-ordinates
of a point Η situated at a distance r from A; we have
2 r
ax + by + ox » R cos -jr
and In this wey, the equation Is determined. We can make It homogeneous by
squaring and taking Into account that the co-ordinates utilised are normal;
we thus obtain
(ax * by + ob)2 - cos2 j[ (x2 + y2 + x2)(a2 + b2 + a2) .
We can see that every circle is represented in the elliptia plane by a conic
bi~tangent to the absolute; the straight line contact Is the straight line
ax + by + ax ■ 0, that Is to say, the polar of the centre of the circle, as
much also with respect to the absolute as with the circle Itself.
2
124. Let us finally seek the analytic expression of the ds of the
elliptic plane. We shall make the central projection of the sphere onto the
plane (P) tangent at one of Its points A and we shall analytically define
a point Η of the plane (P) by Its rectangular co-ordinates X,Y with
respect to two axes with origin A. Let then da « /dX2 + <fY2 be the
ordinary distance between two inflnlteslmally near points Η and N of the
plane (P) (Fig. 12), let α-j be the angle at the centre HON.
Lastly, let us denote by Φ the angle ONN. We have
MN я ОН
α sin φ '
hence
0Ν·ΗΝ sin φ » α· ОН · ON
- £ (Χ2 + У2 + R2)

Ί32
Hen-Euclidean Geometries
The product ON* MN sin φ Is equal to twice the area of the triangle ONN
or alternatively. Is the magnitude of the Μ vector dete mined by the two
vectors OH and MN, the projections of these two vectors being respectively
X
dX
У
cCi
consequently
ON · m sin φ = Α*Μ - VdX)2 + R2(ciX2 + cCi2)
Thus we obtain
л2 ж R2 , R2(dX2 * df2) * (XdV - VdX)Z
(X2 + У2 + R2)
Let us put -г = К ; the above formula could be written as
R*
ds
2 и dX + di2 + K(XdY - YdX)2
[1 + K(X2 + YZ)]Z
W
This brings to light the property already described that the Euclidean metric
of the plane (P) (da2 » ciX2 + rfY2) Is osculatory at A to the metric of
the elliptic plane. The line element obtained Is thus defined by the double
property that the straight lines are represented by linear equations In X
and Y, and that the Euclidean plane with rectangular co-ordinates (X,Y) Is
osculatory to the elliptic plane at the origin of the co-ordinates. It Is
just as well to note that the line element determined applies to all the
elliptic plane tHth the exception of the points situated on the polar straight line
of the point X s Υ * 0 (which corresponds to an Infinity of values of X and Y).

Non-Euclidean Geometries
133
III. TWO DIMENSIONAL HYPERBOLIC GEOMETRY
125. The formulae representing the geometric properties of the figures
outlined In an elliptic plane with curvature -y contain a positive param-
1 R 1
eter К - -V . If we give to this parameter a negative value К * —у , we
R* R*
obtain the hyperbolic geometry. We could directly define It by taking, In the
projective plane, a real conic Γ (the absolute) and calling the dietanae
(Cayleylan or non-Euclidean) between two points Μ and M' of the plane
D
the product by j of the logarithm of the cross-ratio formed by the two given
points and the two points where the straight line joining them cuts the
absolute. If we require this distance to be real for all the segments of the
straight line stemming from M, It Is necessary (and sufficient) that this
point Is situated In the interior of the absolute. The hyperbolic plane is
therefore the manifold formed by the interior points of the absolute Γ. When
the point M' tends towards a point of the absolute, the point Μ staying
fixed, we can see straight away that the (Cayleylan) distance between the two
points extends Indefinitely. The absolute Is therefore the locus of points
at Infinity.
The angle between two straight lines that stem from a point A Is, as
In the elliptic plane, the quotient by 2t of the logarithm of the cross ratio
formed by the two given straight lines and the two tangents taken from the
point A to the absolute. In particular, two straight lines are
perpendicular (In the Cayleylan sense) If they are conjugate with respect to the
absolute, that Is to say, If one of them passes through the pole of the other (a
pole which actually does not exist In the hyperbolic plane proper).
The geodesies of the hyperbolic plane are evidently the straight lines,
for the calculation conducive to these geodesies Is the same as that which
would be made In the elliptic plane (with the sole difference that the
positive parameter К is, here, negative).
The Euclidean postulate Is not true for hyperbolic geometry; for through
a point A situated away from a straight line 0, we could take an Infinity
of straight lines that do not meet 0. Amongst these, two are extremes; they
are the two Lobataheusky parallels, which join the point A to the two points
6 and С of the Intersection of 0 with the absolute (Fig. 13).
u

134
Bon-Euclidean Geometries
126. We have seen above that the perpendiculars raised erected to the
different points of a straight line 0 forn, in the entire projective plane,
a bundle of straight lines whose vertex 1s the pole Η of the straight Hne
0 with respect to the absolute. The straight line 0 Is therefore an
orthogonal trajectory (In the non-Euclidean sense) of this bundle. It 1s easy to
have others. Let us consider In effect (Fig. 14) a conic (С) Ыtangent to
the absolute at the points A and В where the straight line D cuts (Г).
Let Η be a point on this conic; the tangent at Η passes through the pole
N of the straight line HM with respect to the conic (C); a pole which Is
on the straight line AB and which Is also the pole of HM with respect to
(Γ). The tangent HN at (C) Is therefore (from the non-Euclidean point of
view) normal to the radius HM of the bundle. The conic (C) Is thus an
orthogonal trajectory to the straight lines perpendicular to AB.
Fi| ■ ι
On the other hand, we know (see No. 95) that If In any two dimensional
Rlemannlan space we can take a constant length on the normal geodesies with
a fixed geodesic, then the locus of points so obtained Is normal to all these
geodesies. Consequently· the conic (C) is the locum of points obtained by
taking on the perpendiculars to the straight line 0 a constant length, we
call It the line of equal dietana*> or equidistance or even a hyperayoU.
If Instead of taking In the projective plane, a bundle of straight lines
with a vertex to the absolute, or take a bundle of straight lines stemming from
a point A Inside the absolute, the orthogonal trajectories are evidently the
non-Euclidean circumferences with centre A. Consequently, the mm-Buolidean
oiroumferencee are, in the hyperbolic plane, the bitangent aonioe to the
absolute, the straight line of contact being outside the absolute, and the pole of
this straight line of contact is the centre of the oiroieeference.
An Intermediate case Is that of a straight line bundle, having for a
vertex a point Η of the absolute. The orthogonal trajectories are the conies
admitting at Η a third order contact with the absolute; we call these the

Non-Euolidtan Geometries
135
horocyalee; they are therefore at the basis of the orthogonal traj'eotories of
a family of LobaUtheweky parallels.
If we Imagine a point A and a straight line passing through A, the
Μ tangent conies to the absolute which pass through A and are tangent to
the straight line 0 at A contain three categories of distinct curves:
some circumferences, two horocycles and some hypercycles.
To take account of the way In which the curvature of these curves vary,
let us make the representation of the hyperbolic geometry on the plane In such
a way that in ordinary reatanguUzr ao-ordinates the equation of the absolute
Is
X2 + У2 + R2 - 0 ;
at the origin A of the co-ordinates, the Euclidean metric of the plane Is
osculatory to Its Cayleylan (hyperbolic) metric. The absolute Is a circle of
radius R (Fig. 15). If we see It as a horocycle passing through A and
touching Г at B, the radius of curvature (In the ordinary sense) of this
conic at В Is the sane as at A; consequently, the radius of curvature at
A Is equal to R. The end result Is that every horooyale пав a radiue of
curvature . We can then see that every airoumferenae пае a radiue of
curvature leee than R and every hyperoyole a radiue of curvature greater
than R. bf ι*
Besides, we know by analogy with what we had seen In elliptic geometry that
the ordinary curvature of the circles with centre A Is equal to their non-
Euclidean curvature; It Is therefore always greater than «■ .
We arrive at the same result by taking the formula
ρ ■ R tan £ ,
which gives the radius of geodesic curvature ρ of a circle traced on the
sphere and with radius (taken on the surface of the sphere) equal to r. If we
pass from elliptic geometry to hyperbolic geometry, the formula becomes
0 ■ R tanh ~ ;
It snows that ρ goes from zero to R when r Increases from zero to +~.
In the same way we obtain the curvature of the equidistances by noting that.

136
Non-Euclidean Geometries
on the sphere, the locus circle of points situated at a distance a from a
_n
great circle has for radius -y- - a and for the radius of geodesic curvature
ρ - R cot |
In hyperbolic geometry, the radius of curvature of the locus of points situated
at a distance a from a fixed straight line Is therefore
ρ = R coth | ;
It Is always greater than R.
127. The formula which gives the area of a spherical triangle
stays true in hyperbolic geometry, where the sum of the angles of a triangle
Is less than two right angles, and the difference π - (A + B + C) Is equal
с 2
to -y . The area of a triangle could not therefore exceed the value nR .
R*
This limit Is attained for a triangle whose three vertices are on the
absolute; the three sides of this triangle are palrwlse parallels (of Lobatcnew-
sky).
128. The calculation of the line element of the hyperbolic plane leads
to a result similar to that for the elliptic plane. If U 1s adapted to the
plane with (homogeneous) projective co-ordinates, such that the equation of
the absolute Is
and If we assume that the point» Inside the absolute leave Г negative (as
Is always permitted), then we shall have
ds2 - F(drtdy,dz)
with the constraints on x, у, я being
F(xti/,a) - -RZ - 1 .
We might also note the form already Indicated In No. 124
[1 + K(X' + У*)Г

fion-Euolidean Geometries
137
IV. CONFORHAL REPRESENTATION OF SPHERICAL AND HYPERBOLIC GEOMETRIES
129. In the preceding section we utilised the geodeeia representations
of the spherical elliptic and hyperbolic geometries, a representation In which
the straight lines have straight lines for their Images. Another
representation which Is just as Important Is that preserving the angles.
This can easily come about» In spherical geometry» by making a stereo-
graphic projection of the sphere onto a plane; It Is not therefore necessary
to consider this plane as a projective plane, but as the plane In the theory
of functions that possesses a single point at Infinity (that which corresponds
to the pole Η of the stereographlc projection on the sphere). With
preference we shall take for the plane of projection the tangent plane to the
sphere at the point A diametrically opposite the point H. We shall thus
have a Euclidean osculating plane at A to the two dimensional Riemannian
space constituted by the surface of the sphere.
In this representation of spherical geometry» a circle Is represented by
a circle, a "straight" line (great circle) likewise by a circle, but enjoying
a characteristic property, 1-е. that the influence of the point A iHth
reepeot to this circle ie constant and ie -4R . We have, by studying the
diagram In rig. 16,
AH * 2R tan HHA
whence
AN - 2R tan NHA
ДМ-ДЙ - - 4R'
t-i*. in.
ЧМ
We could even say that the "straight" lines are represented by the circles
orthogonal to the (Imaginary) circle with centre and radius 2tR; this circle
takes the name the absolute\ Its equation Is, In rectangular co-ordinates,

138
Non-Suolidean Geometries
X2 + Υ2 + 4R2 - О
It Is the Intersection of the plane (P) with the i sot ropy cone of vertex H.
The line element of the sphere» with rectangular co-ordinates X.Y of
the stereographlc projection, Is easy to determine. If In effect, И and Ρ
are two Inflnlteslmally close points of the plane, arising out of two points
M\ P* of the sphere (Fig. 16), we have
HP HH-HP
/dV? + <fYZ , X2 + Y2 f 4R2 . , Л /v2 . »2
3i 4R
2 1 +^(X< + Y<)
? ?
dz ■ о χ я—* ; (5)
[1 ♦ f (X2 ♦ Y2)]2
2 2 2
we can verify that the Euclidean dz \ dX + di Is osculatory at A to the
о
spherical dz .
130. To define directly the distance between two points Μ and Ρ by
means of the absolute in the oonformal plane, we consider the circle
orthogonal to the absolute passing through Η and P, and let Q. and Q- be the
two points (Imaginary conjugates) where It meets the absolute. The four points
M. P. Qi. Q2· being situated on a similar circle, admit on this circle a
certain cross ratio. To evalute It, let us observe that these four points arise
out of the projection of four points M', P', QJ, Q' of a great circle of the
sphere. Now the absolute Is the trace, on the plane of projection, of the
isotropy cone with vertex H, a cone which meets the sphere following the
umbilical. The points QJ and Q' are therefore the two cyclic points at
Infinity I and J of the great circle M'P' of the sphere. If then we
take any point K' on this great circle (Fig. 17), the cross ratio of the
four straight lines K'M\ K'P', ΚΊ, K'J, Is equal to *2t6, on denoting
by θ the angle M'K'P', or even to ei<4/R, where d denotes the distance,
taken on the sphere, of the two points M' ,P. We then have, on the plane of
projection, for the (spherical) distance between two points M,P, the formula
d - % log (HPQ^) (6)

Bon-Euolidean Geometries
139
131. To obtain а с on formal representation of hyperbolic geomtry, let us
take the absolute (r) In a plane (τ). There exists an Infinity of points
Η such that the cone with vertex Η and base (Γ) Is one of revolution (In
the ordinary sense of the word). Let us take one of these points and Inscribe
In the cone a sphere (Σ) (Fig. 18): the cone touches It and so describes
a circle (Γ) which divides the sphere Into two "caps." This established,
we shall let correspond to every point Η (Inside the absolute) In the plane
(it), the point M' where the straight line HM meets one of the two caps,
chosen once for all. We thus obtain a representation of the hyperbolic plane
on the cap of the sphere In question; the points of (Γ) projecting onto
(Γ), the circle (Γ) will then be said to be the absolute.

140
tkm-Euolidean Geometries
Let us prove that this representation preserves angles, that Is to say
the (ordinary) angle at which two curves fro» M' Intersect on the sphere Is
equal to the (non-Euclidean) angle at which the corresponding curves Intersect
In the plane (π). Let MT, HTj be In the plane (π); MT, M'TJ In the
tangent plane to the sphere at M' are tangents to the curves In question.
The non-Euclidean angle THT. depends on the cross-ratio of the straight lines
MT, MTj and the two tangents taken fro» Η to (Γ). This cross ratio Is the
same as that for the two lines MT, M'Tj and the two tangents taken fro»
M' to the sectional curve of the cone with vertex Η through the tangent
plane at Η to the sphere. Now. following Dandelln's theorem, this section
Is a conic In which M' Is one of the foci; consequently the two tangents
stemming from M' are the two ordinary, Isotropic straight lines stowing
from M' In the tangent plane. Finally, the ordinary angle T'MTi Is equal
to the non-tuclldean angle ΪΗΤ*. This is what we wanted to prove.
132. In the conformal representation In question, a straight line of the
hyperbolic plane has for Its Image the section of the sphere through a plane
passing through H; this section Is л circle orthogonal to the absolute (Г)
to the points Qj and Qi, projections of the points Q, and (L where the
straight line meets the absolute. If we take two points M' and P' on this
circle, providing two points Η and Ρ of the straight line considered In
the hyperbolic plane, we can propose to evaluate the (non-Euclidean) between
these two points In terms of the cross ratio (H'P'Qj.QI) of the two given
points M\ P' and the two points QJ, Q' (Fig. 18). Now we have (see No.
125)
d = | log (MPQ^) - \ log (H · M'P'QjQJ) ;
the question here Is to compare the cross-ratio (M'P'QjQ') with the cross-
ratio (H-M'P'Q|Q2).
To make this comparison, let us make a nomographic transformation which
sends the points Qj and Q' to the two cyclic points at Infinity; the
point Η then becomes the centre of the circle (Fig. 19) and we have
(H-M'P'Q^) ■ e2ia
where a denotes the angle M'HP'. On the other hand. If K' Is any point
on the circumference, we have
(M'P'Q^) - (K' · M'P'Q^Q^) * β2χβ
where β denotes the angle H'K'P'. The result Is that
(H-M'P'QjQ^) ■ (M'P'QjQ^)2 ,

Νοη-Euolidean Geometries 141
and consequently
(б1) d- flog (H-M'P'QjQ£)
- R log (M'P'QjQ^) ;
this fomula га the companion to the formula in (6), found in spherical
geometry.
133. It Is now easy to see that the (non-Euclidean) circumferences, the
horocycles and the hypercycles of the plane (π) are represented on the sphere
by the circumferences (or the arcs of the circumferences). In fact, the cone
with vertex H, and having for Its base one of these curves, Is therefore
Ыtangent to the sphere; consequently its intersection ьп-th the sphere dsoom-
poses into two plane curves, I.e. two circles. If we start from a
circumference of the plane (w), we obtain on the sphere two circles, but only one of
them Is completely Inside the relevant cap. If on the other hand, we start
from an equl-distance, we obtain two circles which meet on (Γ1) and It Is
only necessary to preserve the arcs situated Inside the cap. These two
circular arcs correspond to the two parts of the equidistance separated by Its
points of contact with the absolute.
134. We can now move onto a conformal representation of the hyperbolic
plane on the ordinary plane by making an Inversion having Its pole at a
point of the sphere. If this pole Is taken Inside the other cap, the points
of the hyperbolic plane are represented by points Inside a certain real
circle {the absolute), the "straight lines" being represented by the circles
orthogonal to the absolute.
A more Interesting representation, and one used by H. Poincare In his
theory of Fuschlan functions, consists In making an Inversion whereby the pole
Is on the absolute (Γ). This absolute thus becomes a straight line and the
points of the hyperbolic plane are represented by the points of one of the two
half-planes, bounded by the straight line {the Poincare half-plane). If we

142
lion-Euclidean Geometries
denote this straight line (absolute) by Δ, then the "straight lines" are
represented by the semi-circles having their (ordinary) on Δ and are
situated In the Polncare half-plane.
We could easily recover the property of the non-Euclidean circles by
representing them as the circumferences. Let us, In effect, consider the bundle
of orthogonal circles at Δ, and passing through a point A of the Polncare
half-plane (and the point A' symmetric to A, with respect to Δ). The
orthogonal trajectories of this bundle will represent the non-Euclidean
circumferences whose (non-Euclidean) centre Is at A; we know that they form a
bundle of circles, of which A and A* are the limit points (the Poneelet
points) (Fig. 20).
.**-..
Let us now take the bundle of circles (parallel, non-Euclidean straight
lines) passing through a fixed point A of Δ and having their centres on Δ:
their orthogonal trajectories from another bundle of circles· tangent to A
at Δ' (Fig. 21); they represent the normal horocycles to the parallel straight
lines in question.
Flf. ii.

Ban-Euolidean Geometries
143
Let us now take a bundle of circles having their centres on Δ and
admitting two points bounded by A and В on Δ (Fig. 22); their orthogonal
trajectories are the circles passing through A and B, which thus represent
the hypercycles; amongst then we find a (non-Euclidean) straight line
represented by the half-circle with diameter AB. The (non-Euclidean) straight line
segments ΗΝ, M.N,» MJL» are perpendicular to the (non-Euclidean) stralqht
line AB of constant length.
\
135. If we take the Polncare* half-plane with respect to a system of
rectangular co-ordinates, the straight line Δ being taken as the axis OX· we
can easily find the analytic expression for the line element of the hyperbolic
plane. Let us consider. In effect, two neighbouring points И and N, and
the circumference orthogonal to Δ passing through И and N (Fig. 23).
Let A and В be the points where It cuts Δ.
We could rationally express the co-ordinates of a point Η of the
circumference by means of the parameter
t ■ tan θ
where θ denotes the angle ВАИ. The values of t correspond to the points

144
Hon-Euclidean Слоте trie в
Μ, Ν, Α, В
are respectively
tan θ, tanfe + de), », 0,
and we have
(imka\ tan(9 + dQ) m , , d tan θ
lMKAB) tan θ ' + Un θ
. 1 + 2d9
1 +Ш1в
consequently
A ■ " log (1 ♦ ^ЙТ»)
κΙΤΤΓΤθ *
Let r, therefore, be the ordinary radius of the circle AHNB, let В be Its
centre; we have
ψθ) m r d(Ze) я
sin 2Θ r sin 2Θ
and consequently
л2 . R2 dX2 + di2 .
/&X2 + cfY2
Υ
1 dX2 + di2
К у2
(7)
136. In sureting up, we have determined the Important line element for
elliptic geometry,
ds2 - d*2 * *2 УС" -™)2 . (4)
[l + κ(χζ + γ£)Γ
The co-ordinates X,Y are chosen In a way that every straight line Is
represented by a first order equation. All the points of the space are analytically
represented by means of these co-ordinates, with the exception of points on one
line, i.e. the polar line of the point (X - Y- 0).
Spherical geometry admits л line element
a2. *\*f ; (5)
[l + J (xz + Yz)r
the co-ordinates X,Y are chosen In a way that realises а сonfо пи 1
representation on the Euclidean plane; all the points are analytically represented by

Hon-Euclidean Geometries
145
means of these co-ordinates, with the exception of just one, the antlpode of
the point (X-Y-0).
Hyperbolic geometry admits three Important line elements. I.e. two of them
are obtained by giving, 1n the preceding formulae, a negative value to K; 1n
the other case, all the points of the hyperbolic plane are represented
analytically by means of the co-ordinates X,Y constrained to satisfy, 1n the first
case, the Inequality
1 + K(X2 + Y2) > 0 ;
1n the second case, the Inequality
1 + | (X2 + Y2) > 0 .
Moreover, there exists an Important line element (conformal representation on
the Polncare* half-plane). I.e.
de я - τ —ρ— ; (?)
all the points are analytically represented by means of the co-ordinates X,Y
(with the condition Υ > 0). The (non-Euclidean) element of area is
d - R2 &4L , (B)
and It Is easy to see that a triangle whose three vertices are at Infinity on
the absolute, such as that Indicated In Fig. 24, and which Is bounded by a
semi-circle with centre 0, (ordinary) radius a, and between two straight
lines AC and ВС, has an area ttR . If we have 0Α-0Β-α, we have to
calculate the Integral
f
I - R2//^fL
г
across the domain defined by the Inequalities
2 2 2
Γ + Γ 2: a
the integration gives
!-"7 77*Τ'"'2 ■

146 Non-Euolidean Geometries
We can see quite clearly In the diagram that the three angles of this triangle
are zero.
V. THE DISPLACEMENT GROUP DF THE NON-EUCLIDEAN GEOMETRIES
137. Each of the three, two-dimensional non-Euclidean spaces (spherical,
elliptic, and hyperbolic) satisfy the axiom of free mobility and admit a
continuous family of direct Isometrles {non-Euolidean displacements) depending
on three parameters; It also admits a three parameter family of Inverse
Isometrles.
Let us put aside the case of the sphere, which Is well known, as well as
that of the elliptic plane and only concern ourselves with hyperbolic
displacements.
If we make a geodesic representation of the non-Euclidean plane (of
Lobatchewsky) on the projective plane, we Immediately see that every homography
whiah preserves the absolute represents a non-Euolidean isone try. It Is not,
however, unnecessary to point out that the homography preserves the set of
points Inside the absolute, since these points are characterised by the
Invariant projective property that the tangents taken from one of these points to the
absolute are Imaginary.
We can, as we know, express the homogeneous co-ordinates of a point of the
absolute conic as real, rational functions of a real parameter t\ every
homography preserving the absolute will establish a holography on the parameter
t. Conversely, every homography on the absolute Implies a tomography In all of
the plane; this pertains to the fact that point of the plane Is completely
defined by the contact points of the tangents steaming from this point to the
absolute and every "straight line" of the plane through Its points of
Intersection with the absolute. This Is also evident by the calculation. If we
prefer to do so and this Is always possible, of the projective co-ordinate system
of the plane, under the condition that we have on the absolute,

Bon-Euolidean Geometries
147
7 * T
If ме take on the absolute the tonography
*. . at + b
we will deduce. In the plane Itself,
2 2
par1 ■ ax + laby + b *
ρ»' ■ αα'χ + (ab1 + Ъа')у + Ы>'*
2 2
ρ*' ■ a' * + 2а*Ь'у + b' *
Therefore, In order to make а classification of the non-Euclidean displacements,
we have to нке a classification of the real nomographics In one variable. We
will always have at first a large division, following our assumption of
ab' - ab1 to be positive or negative. The first case corresponds to
displacements proper, the second case, to displacements following a symmetry.
138. The displacements proper will be classed by the nature of the double
points of the homography established on the absolute.
1 . The double points are imaginary.
The displacement here leaves Invariant the real straight line joining the two
double points (a straight line outside the absolute) as well as Its pole A
(Inside the absolute). The displacement so obtained Is evidently a rotation
about the fixed point A. In a continuous rotation about A, the different
points of the plane describe (non-Euclidean) circumferences centred at A.
2 . The double points are real and distinct.
If A and В are these two double points, the displacement leaves the straight
line AB Invariant, each point of this line being displaced along a segaent of
constant (non-Euclidean) length. We have what might be called a (non-Euclidean)
trmeUition. In a continuous translation of axis AB, the points which are not
on the axis, do not describe straight lines, but describe hypercycles (with
constant curvature less than jr).
3 . The double points are identified.
If A Is the unique double point, the corresponding displacement leaves the
point A fixed and consequently transforms the straight lines between
themselves In the bundle of the Lobatchewsky parallels with vertex A. In a

148
Son-Euclidean Geometries
continuous displacement about At each point of the plane describes a horo-
cycle represented by a conic osculating the absolute at A. The displacements
following a symmetry could only occur If the homography of the conic has real
and distinct double points. The equation giving the double points Is
a't + {bl -a)t - b - 0 ;
Its discriminant Is
{bl -a)2 + Aha1 = {bl + a)2-4(ab' -bal) > 0 .
If A and В are the double points of the homography, the corresponding
transformation results from a translation with axis AB followed by a symmetry
with respect to AB.
139. Let us now move onto the сonformal representation and to simplify
matters, let us take the representation on the Poincare half-plane. Following
the definition obtained for the non-Euclidean distance between two points,
every transformation that will alter the circles on preserving the absolute Δ
will be an isometry. Now, In the oonformal plane, there exists two classes of
point transformations taking circles to circles (the Kreieveraandeohaften of
MBbius). Let us represent a point of the plane by its affix « ■ X + ΐΥ with
respect to two rectangular co-ordinate axes (we shall take Δ as our real
axis). The first class of transformations (direct transformations) Is defined
by the formula
1 α'* + β'
where α, β, α', β1 are arbitrary aorrplex constants. The second class
(Inverse transformations) Is defined by
axQ + β
"' " α'*0 + β1 '
where ж denotes the quantity X - ΐΥ, the complex conjugate of я.
The transformations of the first class which leave the real axis Invariant,
and which preserve eaah of the half-planee bounded by Δ, are obtained by
giving to α, β, α1, β', the real values a, bt a', b':
*' * πττ* {ab' ·Ьа' > 0) · (9)
The corresponding non-Euclidean displacements are again classified by the
nature of the double points of the homography (9). If the double points are
Imaginary (conjugates), they are the affixes of a point A of the Polncare"

Son-Sualidean Geometries
149
half-plane and of the point A' symmetric with respect to Δ. The
displacement Is a (non-Euclidean) rotation about A.
If the double points are real, then they are the affixes of two points A
and В situated on the absolute. The corresponding displacement Is a transit
tlon having the semi-circle (non-Euclidean "straight line") described on AB
as diameter, as the axis.
If then the double points are Identified, we obtain on the absolute a
point A and we have on It a displacement transforming the (non-Euclidean)
straight lines, that Is, the circles orthogonal to Δ and passing through A.
If In particular the point A Is taken to Infinity, then the corresponding
bundle of Lobatchewsky parallels Is represented by the ordinary bundle of
parallels at OY: the horocycles which are orthogonal trajectories are
represented by the parallels at OX, and the corresponding non-Euclidean
displacement Is represented by an ordinary translation parallel to OX.
Let us move onto displacements followed by synmetry; they are analytically
represented by relations of the form
a* + b
(10)
where the coefficients a, b, a', bl, are real and abl - bal < 0 (so that
the two half-planes are preserved by the transformation). Such a
transformation leaves two real points of the absolute Invariant, and we recover the
Interpretation described above: a translation followed by a symmetry with
respect to the axis of the translation.
VI. THREE DIMENSIONAL, NON-EUCLIDEAN SPACES: PROJECTIVE REPRESENTATION
140. As we did for n*2, we could start with a spherical space In which
each point Is defined by four co-ordinates x, y> z9 t satisfying the relation
x2 + y2 + a2 + t2 = R2 ,
the line element being
da ■ dx + dy + dz + dt
Such a space Is said to have curvature -y .
R*
Let us take a projective point of view of things. In three dimensional
projective space, with respect to the projective co-ordinates x, y> a, t,
let us take an absolute quadratic with equation
f{x,y,z,t) * 0

150
Son-Suolidean Geometries
Let us restrict our attention to multiplying the projective co-ordinates of
a point by a factor such that we have
/(x.*.«,0 ■ £ (Π)
where К Is a given constant, and let us define the line element
da1 ■ f{dx,dy,da9dt) . (12)
In order that the Rlemannlan space, so defined, has real points and that Its
line element 1s a definite positive form, then it is at first necessary that
the polar plane of the point (r,y,*,fc) with respect to the absolute does
not cut the absolute; In effect, the four quantities dr, dy, de, at are
subjected to the unique relation
fjb + Гуау + f]da + ftdt - 0 ,
which says that the point {dx>dy>da,dt) Is In the polar plane of the point
{x,y,u,t). If the polar plane did cut the absolute, then the differential form
2
da would not retain a constant.
The preceding condition instantly excludes the regular quaarics* for every
plane cuts such a quadrlc, following a real curve.
There remains, therefore, just two possible hypotheses.
I. The absolute is an imaginary quadric (\jith real equation). It Is then
necessary for the form / to be positive definite; consequently the constant
К Is necessarily positive. We obtain the elliptic space with positive
curvature K.
II. The absolute is a real, irregular quadric, e.g. an ellipsoid. They are
planes outside the quadrlc that do not cut It; therefore f must be positive
outside the quadrlc. The points {x,y,M9t) of the Rlemannlan space being
Inside the quadrlc (such that their polar planes are outside), the constant К
Is necessarily negative. We obtain the hyperbolic space vith negative
curvature K. The form / Is decomposable Into three positive squared terns and
one negative.
141. We could directly obtain the projective representation of the
elementary distance between two points M,M'. Every point of the straight line
NM' has co-ordinates of the form
χ + \dx , у + Щ Λ ι + λώ , t + Xdt ;
the values of λ correspond to Its points of Intersection, P, and P„, with
the absolute ere given by the equation

tbn-Euclidean Geometries
151
fix + Xdxt...tt + \dt) Ξ j + λ2ώ2 - 0 ,
whence
λι и 7ГБ » λ2= " Ιώ
The cross ratio (ΗΜ'Ρ,Ρ-) Is equal to the cross ratio of the four values
of λ corresponding to the four points In question
(HM'PlP2). (0.1,νλ2>-ψΐ-φ
u *2 . 1 ♦ dett
1 -γ- 1 - ώ/Π?
λ1
on taking logarithms and restricting matters to the principal parts, we have
da ■ —]— log (Η,Η',Ρ,,Ρ-) .
2Л£ ' &
We deduce by Integration, along a finite straight line segment, the expression
for the non-Euclidean distance d between two points Η and Μ', α dietanoe
computed along the straight line,
d ■-!- log (HM'P.PJ (13)
2Л( ' &
whence, by a calculation similar to that made In No. 122, the formula
¥*lfx + yify + *tfm + tift] " l cos {d/Z* ] ' ίΊ4)
We shall see later (In No. 145) the direct proof of the property of the
"straight line" being a geodesic.
142. In the preceding section we subjected the projective co-ordinates
of a point to satisfy the relation f " ν · № could more generally consider
a set of four co-ordinates (x.i/.a.t) that are not all zero, and call this
set an analytic point. Two analytic pointe In which the four co-ordinates
will be proportional but not equal will be regarded as distinct, while occupy-
In" the same position In the space. We shall call the quantity /(*.i/.a,t)
the scalar square of such a point, and more generally, we will call the eaatar
product of two analytic points (х,у,я,£), (х',y* ,я*,fc*), the quantity
*<■■{£♦*■£♦■■{?♦*·$ ·

If M and N denote any two analytic points, the squared scalar of the point
λΜ + μΝ, whose co-ordinates are deduced from those of И and N by
multiplication by μ and Λ respectively and then addition Is equal to
xV + 2λμΗΝ + μ2Ν2
Two points whose scalar product Is zero are conjugates of each other with
respect to the absolute; each of them Is In the polar plane of the other.
The Infinitesimal vector defined by two Inflnlteslmally near points Μ and
И' Is analytically defined by the four quantities dx, dyy ал, dt. Let us
constrain the co-ordinates of the points In question to the condition
f{x,y,a,t) ■ j (I.e. let us assume *r ■ γ); the four numbers dx, dy, ал,
dt, represent an analytic point of the projective space which Is situated In
the polar plane of Μ with respect to the absolute, and whose squared scalar
f{dx,dy,dz,dt) ■ ал Is equal to the squared length of the vector. We will
call It the representative point of the vector.
In a general manner, every vector stemming from a point Μ fn elliptic
or hyperbolic space will be represented by an analytic point situated In the
polar plane of Μ with respect to the absolute, and having for the squared
scalar the square of the length of the vector. This representative point Is,
furthermore, on the straight line from Μ In the direction of the vector.
We can easily verify that the scalar product of two vectors stemming from Μ
is equal to the scalar product of their representative points.
143. The co-ordinates considered up to the present are arbitrary
projective co-ordinates. This generalises the rectangular co-ordinates of Euclidean
space.
Let us start with any point A (with squared scalar γ) In elliptic or
hyperbolic space, and consider the representative points e, e« e. of three
unit rectangular vectors stemming from A. Every analytic point Μ of
projective space will put In the form
Μ ■ tA + xe, + ye~ + ее. ,
and by virtue of the evident relations
2
A· e. ■ 0 e. = 1 e. · e. ■ 0
г г г j
(ΐ t J\ i,U ■ 1.2,3)
we will have the following expression for the squared scalar of И:
m2= 2a 2 a 2.1^2
If the point Μ Is a point of elliptic or hyperbolic space with squared

Яоп-Euolidaan Geometries
153
scalar -»:
J A 2 . 2 . 1 Л 1
ме will have for the line element of the space,
da1 ■ dH2 ■ <ic2 + dy2 + ώ2 + ^ dt2 .
The co-ordinates thus obtained, and bound by the relation
J A 2 . 2 . 1 „2 1
x+t/ +»+■££■■£ »
bear the title, Weieretraae oo-ordinatee, a name given to them by W. Killing.
If ме let К tend to zero, the co-ordinate t takes the value 1, and
2 2 2 2
the de , as we can easily see by the limiting, reduces to dx + dy + da .
The Welerstrass co-ordinates are only defined to within a sign. If In
elliptic space we agree to regard two points whose Welerstrass co-ordinates
are equal and opposite, as dietinott then we should obtain the spherical
space. In the hyperbolic case we could equally regard as distinct the two
points (x,j/,s,t) and (-χ,-#·-3,-έ), but It Is Impossible to pass
continuously from a point with co-ordinate positive t to a point with negative
co-ordinate t because of the relation
tZ ■ 1 - K(x2 + у2 + я2) 2 1 .
The convention which takes elliptic space to spherical space would here lead
to dieoonmoted space, that Is with two manifolds completely separated from
each other. We exclude one such possibility. We can see. In the meantime,
on account of what has preceded, that we can always assume the co-ordinate
t to be positive In hyperbolic space.
144. The passage from one system of Welerstrass co-ordinates to
another Is made by means of a linear transformation. If the first system Is
fixed and the second varies, we obtain the most general linear transformation
2 2 2 12
leaving the form χ + у + я * γ t Invariant. All these transformations
also leave Invariant the line element
dx + dy + da+jrdt ;
they therefore define the group of isometries of the space. This Is a six
parameter group, the number of arbitrary quantities which enter Into the most
general system of Welerstrass co-ordinates, that Is, on the base of the most
general tetrahedron which Is conjugate with respect to the absolute.

154
Hon-Eualidean Geometriee
In elliptic space, we obtain the same isometry by changing all of the
signs of the coefficients; In contrast, we thus obtain two distinct
displacements In spherical space. In hyperbolic space, the coefficients cannot be
sign-changed If the co-ordinates t and t are constrained to be positive.
We can easily show that the determinant of the coefficients is equal to
±1. The displacements proper correspond to the case where the determinant of
the coefficients of the substitution Is +1; the displacements followed by
symmetry correspond to the case where this determinant Is -1.
145. It is a straightforward matter to determine an osculating Euclidean
metric at a given point A to the metric In elliptic or hyperbolic space.
Let us take In effect a system of Welerstrass co-ordinates i, y, a, t with
origin A. In the neighbourhood of the point A, we have
tZ - 1 - K(x2 + yZ + я2)
t - 1 - \{XZ + y2 + 2t2) + ...
By expressing the line element of the space In terms of the variables x, у, a
alone, and neglecting terms greater than the first order in the coefficients,
2 2 2
It stays simply as dx + dy + dz , which Is a Euclidean line element. We
therefore obtain a representation of the given space on the Euclidean
osculating space at A by representing the point (x,y,a,fc) by the point with
rectangular co-ordinates x, yy a. We have another, more convenient
representation by Introducing the ηοη-honogeneous co-ordinates
we have, to the same degree of approximation as before,
2 2 2 2
d« ■ яг + di& + car .
In the representation on this Euclidean osculating space at At the absolute
Is represented by the аркетв
X2 + Υ2 + Ζ2 + I - 0 .
As a straight line In projective space (In which elliptic or hyperbolic space
Is localised) Is represented In the osculating Euclidean space at one of Its
points A by a straight line, and that this line has zero Euclidean
curvature at A, It results that every straight line In projective space has a
zero (non-Euclidean) curvature at each of Its pofnt and Is therefore a
geodesic In non-Euclidean space. Consequently, the planes of projective space
are totally geodesic surfaces, the axiom of the plane Is thus verified In
elliptic or hyperbolic space.

Ηση-Eualidean Oeometriee
155
We can also see another Important property. Given a straight line
emanating from A, the parallel straight line (In the sense of Levl-Clvlta) emanating
from an inflnlteslmally near point A1 must be represented. In the osculating
Euclidean space at A, by a straight line parallel (In the ordinary sense) to'
the first; consequently. In projective space, it cute the first at a point
t ■ 0 of the plane, t being a polar of the point A with respect to the
absolute. We thus obtain a simple geometric construction of this straight
line.
2
We can easily calculate the da of the space In terms of non-homogeneous
co-ordinates Χ, Υ, Ζ. The Welerstrass co-ordinates are provided by the
equations
tX я у ■ tY , a ■ tl
with
We have
t2
1 + K(X2 + Y2 + Z2)
da2 - t2loH2 + <iY2 + dl2 + 2 Щ- (Ш + idi + Idl) + Q-
t
(X2 + Y2 + Z2 + -p)]
By utilising the relation
dt_ я _ К(Ш + YrfY ♦ Ml)
Ь 1 + K(XZ + YZ + ZZ)
we find, after some simplification
da2 = dX2 + di2 + dl2 + KC(YdZ-ZdY)2 + (ЫХ-ХД)2 + (Xdi - YdX)2]
[1 + K(XZ + YZ + ZZ)]Z
(15)
This formula generalises that of (4) which was established for.two dimensional
spaces.
146. The generalisation of Cartesian co-ordinates naturally comes about
by taking any point A (with squared scalar A) of elliptic or hyperbolic
space and any three vectors stemming from A-, that are represented by any
three analytic points e,, e-. e~ of the polar plane of A, with respect to
the absolute. If we put
Μ ■ tA + xe, + yey * ae- ,
we have

156
fkm-Eualidean Geometries
2 12 2 2 2
Μ « ^ t + 0jj* + ff22j/ + ^33а + 2^23уа + 2д^мх + 2^12a^ ,
where the coefficients д.. denote the scalar products е.* е.. The preceding
Cartesian co-ordinates are adapted to the theory of curvilinear co-ordinates
just as they are for Euclidean space. If we choose any system of co-ordinates
и un In an n-dlraenslonal non-Euclidean space, we will attach to each
point Μ of the space a system of Cartesian co-ordinates defined by Its origin
Η (with squared scalar p-) and the representative points of the vectors
- --SL
t 3u. '
t> ] n
The geometric functions H, e. of u u satisfy the relations
with
da = ЯлА* &t
2
Conversely, knowing the da of elliptic or hyperbolic space permits, as In
Euclidean space, the local reconstruction of this space. We have. In effect,
relations of the form
dM = аиг*.
(16)
The
are
Pfafflan
0
ω.
г
expressions
= Г0. du" ,
гг
determined by taking
M2
ω* - Γ* du*
г гг
account of the
« | , H-e{ ■ 0 ,
relations
ег ' ej B
Я.
t>3
which differentiated, give
»;--"»«** °r An·-*** · <i7)
άβ..·<>3ΐΛ + <>ίΑ·Μ"ίά*"3ί ■ (18)
The Integrablllty condition of the first equation In (16) then gives from which
Γ*.-Γ*. . (19)

Bon-Euclidean Geanetriee
157
We can see that the quantities Π, and the forms ω. are exactly those
which had been determined In the general theory of Rlemannlan spaces. In
particular, we can deduce, for the elementary absolute or aovariant
displacement of the vectors е., the relations
"Ч""*»* *t **«**" · <2°>
147. Let us now seek the conditions that the coefficients д.. must
satisfy In order that a given da Is to be locally elliptic or hyperbolic with
given curvature K. The equations In (16) must be totally Integrable. As
In No. 43, by expressing the conditions of Integrablllty, we get the following
relations which generalise those In (28) of No. 43,
эг* эг*
- -$ ♦ <4Ί, - to - ««fc, - £fa> · <2l>
гг
Ъи8 Эц
а
where ε Is equal to 1 If α* β and 0 If α^ β. We can put them In a
α
condensed form, generalising the equations In (30) of No. 45, i.e.
*4 ' CV£] " "К *«Е*Л*<*] · (22)
VII. THREE DIMENSIONAL NON-EUCLIDEAN SPACES: CONFORMAL REPRESENTATION
148. We can obtain а сonformal representation of three dimensional
non-Euclidean space on ordinary Euclidean space, or rather, on an anallagmatic
space. This space Is none other than Euclidean space, but completed by a
unique point at infinity, Instead of It being completed by an Infinity of
points forming a plane at Infinity, as In projective space. On denoting by
Χ, Υ, Ζ, the ordinary rectangular co-ordinates, every point of an anal lag-
ma tic space Is analytically represented by five homogeneous co-ordinates
χ , χ,, x2» хз» x4> that are not *11 zero, and are defined by the relations
xo s fl ш x2 _ x3 x4
Τ Χ Τ "Τ χ2 + ν2 + ζ2
(23)
these co-ordinates are further bound by the quadratic relation
Ω(χ) = x2 + x2 + x2 - xQx4 = 0 .
The points for which the co-ordinate χ Is non-zero are points at a finite
distance In Euclidean space; If the co-ordinate χ Is zero, the relation
Ω ■ 0 shows that χ,, χ? and

158
!km-Eualidem Geometries
one point whose only non-zero co-ordinate Is x.: It Is the point at
infinity.
Every sphere Is represented by a linear equation
V4 " ^l'l ' Utl ' 2αΛ +Υο'° ; (25)
the coefficients a , a,, a-» a-, a. are said to be homogeneous co-ordinates
of the sphere. If aQ f 0, we have a sphere proper, If α ■ 0, we have a
plane. The planes are therefore regarded as particular spheres, characterised
by containing the point at Infinity: In order that the equation fn (25) Is
verified for χ - x-j ■ x- ■ *3 ■ 0, x- f 0, It Is necessary and sufficient
that a Is to be zero. The linear transformations carried out on the current
о
co-ordinates x. and leaving the form Q(x) Invariant are the analtagmatic
traneformatione. They take spheres (and planes) to spheres (and planes).
These transformations leave the quadratic differential form Q(dx) Invariant.
For χ ■ 1, this form reduces to aft + dY2 + In the general case, a
simple calculation yields
U(dx) - x*(ifX2 + di2 * dlZ) , (26)
consequently the anaUagmatlc transformations are с on formal transformations:
they preserve angles.
We show (LlouvlUe's theorem) that every conform! transformation of
three dimensional space, be It a displacement whether or not accompanied by a
symmetry, a transformation by similitude, (direct or Inverse), or a similar
transformation followed by an Inversion. These transformations decompose Into
two distinct connected families (direct or inverse anaUagmatlc
transformations); they form a 10 parameter group.
149. Let us consider a fixed sphere to whfch we shall assign the role
of being the absolute in the projective representation of non-Euclidean
geometries.
Let us take, for example, the sphere with equation
xo + Τ x4 ' ° ' or X2 + Y2 + Z2 + £ « 0
This sphere Is real If К < 0, Imaginary but with a real equation of К > 0.
This sphere fixed we shall normalise the homogeneous co-ordinates χ.
by the condition
xo+ ί x4"Ί 0Γ roC1 + τ (χ2 + ^+ z2)]"Ί ■ (27)
and will put

Son-Eualidaan Geamvtriea
159
xo-fx4"u :
from It, ме deduce
xf*'4r- ■
The form Ω(χ) could then be written
qm , 4 + 4 + 4 + £ -1 ,
of the type that for the co-ordinates of a point, we have
2 2 2 12 1
xl + x2 + x3 + К u " К '
and the quadratic differential form Is written
Q(dr) - dr2 + dx\ + с&з + \ du2 .
On account of what was said In No. 143, x, y, a could be regarded as the
Weierstrass co-ordinates of a поп-Euclidean space with curvature K, and the
form Q(dx) Is the line element of 1t. Now on account of (26) and the value
of χ given by (27), we have for this differential form the expression
a.2 » d*2 + Л2 + Д2 /or)
[1 + j(X' + Yb + Zc)]'
2
This Is typical of the expressions for the da of the non-Euclidean space
with curvature К In Its conformal representation. This expression, already
Indicated by Riemann for any number of dimensions of the space, Is the general·
isation of formula (5) In No. 129.
In formula (28), Κ > Ο,Χ,Υ,Ζ are the rectangular co-ordinates of any
point of anallagmatlc space although the formula falls to hold for the point
at Infinity. If К Is negative, the point (Χ,Υ,Ζ) Is assumed to be Inside
the absolute; but we could also regard the non-Euclidean space as realised by
the exterior of the absolute Including the point at Infinity. In the first
case the co-ordinate χ Is essentially positive, 1n the second case. It 1s
essentially negative.
2
150. We have another form for the da of non-Euclidean space with
co-ordinate К by taking for the absolute, not a sphere proper, but a plane,
for example the plane Ζ - 0. For this, we shall normalise the homogeneous
χ. by the condition
x, ■ — , whence χΛ ■ —
3 Л? ° Ml

160
Non-Euolidaan Geometries
We then have
2 2 1
n{xj = a:1 + x2 - xQx4 - ^ , (29)
2 1
and the da of the non-Euclidean space with curvature ~ Is the form Q{dx)
which here gives, on account of the value of χ ,
л,2 = -1 M* + <f + aZ . {30)
We can besides, pass from (28) to (30) by an anallagmatlc transformtlon, for
К 2 2 2
example an Inversion, transforming the sphere 1 + j (X + Υ + Ζ ) - 0 In
the plane Z-0, or rather transforming the linear form χ + ^x. to the
linear form Д:,.
With one or the other choice of the absolute, the non-Euclidean planes
are represented analytically by a linear and homogeneous In χ,, х-» хз» u "
χ - j χ. (first case) or In г , χ,, χ-, χ» (second case)
In all cases, we obtain spheres orthogonal to the absolute. The result
1s that the non-Euclidean straight lines are represented by cl rcunferences or
(stralqht lines) orthogonal to the absolute. The non-Euclidean distance
between two points Μ, Η' taken on the circumference (or the straight line)
orthogonal to the absolute which joins them. Is given by the formula
d --Ϊ- log (ΜΜ'Ρ,Ρ,) (31)
/Πζ ' c
where P, and P» are the two points where this circumference cuts the
absolute.
This formula Is the generalisation of the formulae In (6) and (6') of
Nos. 130 and 132, tenable In the conform! representation of the two
dimensional non-Euclidean spaces,
151. The formula in (14) In No. 141 , which Involves the left hand side
of the equation for the absolute In the projective representation also has Its
analogue In the conformal representation In which the polar form of Q(x) Is
Included. We have the relation
bil-i^M-i]
where the x. and x1. are the normalised co-ordinates of two points Μ and
г г
Μ" of the space, and d Is their non-Euclidean distance, or we can have
*l*i ♦-2*2**3*3-? Vi " F*4*; ж ^(dA) -1] ·

Non-Euclidean Geometries
161
For К < 0, let us focus our attention on the conformal representation on the
half-space Ζ > 0. The equation of the non-Euclidean sphere of radius r and
having for Its non-Euclidean centre, the point (a.), could be written as, on
1
noting that a, , ■ - | In normalised co-ordinates,
α1χ1 + а^г + a3 ch(r ЛСЦ - j aQx4 - j afQ - 0 ,
or In non-homogeneous co-ordinates,
X2 + Y2 + Z2 - 2aX - 22>Y - 2ch(r/^)Z + a2 + b2 + e2 - 0 . (32)
Interpreted In the language of elementary geometry, the non-Euclidean sphere
with non-Euclidean centre Α(α,δ,ΰ) and non-Euclidean radius r can be seen
to be the locus of points whose ratio of the distances to the point A and
Its symmetric point A' with respect to the absolute Is constant and equal
to the (r ^r).
The points A and A1 are then the Poncelet points of the bundle of
spheres constituted by the sphere In question and the absolute. This result
Is partly explained straight away If we note that the non-Euclidean straight
lines passing through the non-Euclidean centre of the sphere cut the spheres
of this bundle orthogonally. In other words, the circumferences passing through
A and A1 are all orthogonal to the sphere In question (cf. No. 134).
2
Me have then an anallagmatic form for the da of the Euclidean space by
assigning to each sphere of zero radius (e.g. that which has the origin as
centre) the role of the absolute. We will put
x. s 1, whence x„ = —χ χ к ,
4 ° Χ2 + Υ2 + Ζ2
and will then obtain from
,2 ,2. ,2. ,2 ,, _ rfX2 + diZ + dZ2
da - άχλ + dxz + dx3 - dr0dr4 - (χ2 + γέ + z2) -
The planes now have a linear, homogeneous equation· In χ,, χ?, χ., χ.. These
are represented by spheres passing through the origin; the straight lines are
represented by the circumferences passing through the origin. We obtain a
geometric Interpretation of this form In Euclidean geometry by an Inversion
having the origin as a pole. The classical geometry could In fact be
Interpreted by noting that the point at Infinity In an anallagmatic space plays
the part of the absolute.

162
Mon-Buolidean Gaometrite
VIII. NORMAL RIEMANNIAN SPACE, LOCALLY SPHERICAL OR HYPERBOLIC
152. If the line eleeent of a Rlemannlan space satisfies the conditions
obtained In No. 147 In order for It to be locally elliptic of hyperbolic with
given curvature K, then we can develop on the пол-Euclidean, elliptic or
hyperbolic spice, with curvature K, ewery sufficiently snail portion of this
Rlemannlan space. If the space Is no real and with an everywhere regular metric,
we can reltente the arguments that had been made for Euclidean space (Chap.
III). These arguments rest simply on the following two properties of Euclidean
space: the first Is to admit a group of everywhere regular Isometrles; the
second Is to be eirrply ooimeoted. In the sense that ewery closed contour traced
out In this space can, by a continuous deformation, be reduced to a point.
The first property belongs to spherical space, elliptic space and
hyperbolic space. The second, evidently belongs to hyperbolic space. Vie are going
to see that It also belongs to wphtrioal space.
In effect, let the spherical space, In which each point Is analytically
defined by four numbers χ, yt s, t subjected to satisfy the relation
2 α 2 . 2 . г Л
x+y+*+fc"R
Let us make a stereographlc transformation of this space by putting
1&-X . 1&-' ·
Ш
by squaring and taking Into account the relation that exists between x, yt ж,
t, we deduce
R + t
1ГТТ
Χ" + Υ' + Γ
from which we derive
t - R
χ - R
X2 + Y2 + Z2 - 1
η
Xе + Γ + ζ + 1
2X
-n 2 9
X + Υ + Ζ + 1
(33)
у - R -χ R к
1е + Г + 1е + 1
ж ' R
2Ζ
"7 5 ?
χ* + γ* + V- + 1
Let us regard Χ, Υ, Ζ as the rectangular co-ordinates of a point 1n ordinary
space; to every point of spherical space, with the exception of a single point

Mon-Euolidaan Geometries
163
(χ ■ у ■ ж ■ 0, t ■ R), there corresponds one and only one point of Euclidean
space/ Conversely, to every point of Euclidean space there corresponds one
and only one point of spherical space.
This being the case, let us trace any closed contour In spherical space;
we can assume that It does not pass through the point
(x ■ у я ж ■ 0, t я R)
It has for Its Image, In Euclidean space, a closed contour which could be
reduced to a point by continuous deformation. Consequently, the original
contour given In spherical space could also be reduced to a point by a continuous
deformation. Spherical space Is therefore clearly simply connected; but It
Is not the same for elliptic spice, as we shall be shortly.
1БЗ. The theory of normal, locally Euclidean spaces will therefore
extend to locally spherical spaces and to locally hyperbolic spaces. By the
development of the given Rlemannlan space, be It on spherical or hyperbolic
space, the space (spherical or hyperbolic) will be covered entirely once and
only once. If the given Is simply connected, It will be Identical to spherical
or hyperbolic space. Otherwise It will be represented in one of the two spaces
by a fundamental polyhedral defined by a holonomy group engendered by the
Isometrles of the space, and will be subjected to satisfy the following two
conditions:
1. The group Is discontinuous;
Z. None of Its operations (save the Identity) leave a point of the
space Invariant.
Let us put aside the case of two locally hyperbolic spaces. For я -2,
the determination of those of them that are orientable amounts to determining
that Fuschlan group In which no operation leaves an interior point of the
Polncare* half-plane Invariant, i.e. which Is formed exclusively by parabolic
or hyperbolic transformations; It Is quite possible besides, that the
fundamental polygon happens to have an Infinity of sloes.
154. The case of locally spherical spaces gives rise to some Interesting
theorems. Every point of an (n-l)-dlmenslonal spherical space with curvature 1
(we can always refer to this case) Is defined by η numbers χ,,.,.,χ
ι η
We could agree that the point of spherical space which Is exceptional
corresponds to the point at Infinity In Euclidean space; spherical space therefore
does not differ from the anallagmatlc space of No. 148.
In his book entitled: Einfuhmtnd in die Grundlagen dor Gocmetvio% (Paderbom,
1893), Killing incorrectly asserts the theorem that the group transformations
must be exclusively hyperbolic.

164
/km-Euclidean Geometries
satisfying
χ, + x« + ... + χ ж 1
1 2 η
The group of displacements (whether or not accompanied by a symmetry) Is that
of linear orthogonal transformations In η variables. Let
xi ' aikxk
(ΐ = 1 η)
be such a transformation. The equality
,2 . . .2 2 . .2
* ι + ... + X ■ Xi + ... + X
ι η \ n
leads to the following relations between the coefficients:
,2
(34)
Σα,., 3 1
*ki
Zakiakj = °
(t * Ί,.-.,η),
(ΐ г" /; i,J ■ ι n)
These relations allow us to solve the equations of the transformation with
respect to the χ.» which give
xi ' aHxk
U - 1 я)
(36)
We are going to obtain a noteworthy canonical form for the equations of
the orthogonal transformation In (34). Firstly let us see If there exists a
system of values (x^,...,x') that are not all zero, such that the transformed
values (x!,...x') are deduced from the original values by multiplication by
ι η
same factor λ.
The relations
^£ и а1кхк
for the given λ lead to an n-th degree equation (the characteristic
equation)
all ■ λ
*21
znl
Ί2
a22 " λ
\2
ιλη
г2п
a - λ

Son-Euclidean Geometries
165
A system of values of the χ. that are not all zero defines In n-dlmenslonal
Euclidean space a vector stemming from the origin. The orthogonal
transformation In (34) takes one vector Into another, but with the same length and» In
a more general way. It preserves the scalar product of any two vectors.
Let us assume that the characteristic equation has an Imaginary root λ.
There will be a vector e (necessarily Imaginary) which will be transformed
to the vector λβ:
e' - λβ ;
by replacing each quantity by Its complex conjugate (represented by the lower
Index 0) Into this equation, we will have
eo Aoeo ·
we deduce from It
e · eQ - λλ0 e · eQ ,
whence, since the scalar product e»e Is strictly positive,
u0 - ι .
The root λ is therefore of the form ew. We will also have
e2-A2e2 ,
2
whence, as λ cannot exceed 1,
e2-0 .
Let then
e » e1 + £e2
where the vectors e, and e« are real; we have
el " e2 · el ' e2 a ° ■
The two vectors e,, e? therefore have the same length and are rectangular
to each other. The vector e only defined to within a factor, we can take
the vectors e, and e« to be unit vectors and choose them in Euclidean
space to he the first too basis vectors.
Let us replace the x, and the x'. by the projections (Кг,0,...,0)
and (e ,ге ,0,...,0) of the vectors e and e1 respectively, In the
formulae In (34); we obtain

166
Nan-Euclidean Geometries
α,, ■ COS α , Oji ■ -Sin α , α,, ■ 0,...,α , ■ 0
α22 ■ cos α , α32 я 0,...,αη2 β 0
α·,* " s1n α
The first relations in (35), applied to the coefficients of the transformation
(36) inverse of (34), give us, for 1 - 1,2,
α13"°"···α1η" °
α23β° аЫя °
Finally, the orthogonal substitution takes the form
xj * x1 cos α + x, sin α ,
xi * -x-i Sin α + x« cos α ,
x3 " a33r3 + a34x4 + ·" +a3nxn '
:n ' an3r3 + an4*4 +
+ a x
nn η
(37)
This results in the juxtaposition of two orthogonal transformations, one with
determinant +1, acting on the variables x, and x? alone, the other on the
remaining variables x,,...,x .
If the characteristic equation In λ admits p pairs of Imaginary
roots, we can proceed with the decomposition Into p orthogonal substitutions
with determinant 1, acting on ρ different pairs with co-ordinates
(x,,x2;x3,x4;...;x« i*x?n)» arKi an orthogonal transformation acting on the
remaining n-2p co-ordinates.
The characteristic equation relative to this last transformation now only
admits real roots that can easily be shown to be equal to +1 or -1.
We will then show, as we have just done In several stages, that It Is
possible to decompose the orthogonal transformations acting on χ, +, x^
Into n-Zp transformations, each only Involving one co-ordinate:
χ' « λ χ
α α α
(α - 2ρ+1 η; λα ■ ±1)
The canonical form to which we could reduce every orthogonal transformation Is,
therefore. In rectangular co-ordinates.

Non-Euclidean Geametriee
167
■ ·
(38)
χο· ι ' xo- ι cos a. + x«. sin α
tt-1 Zt-1 г 2г г
χό. ■ -x«. , sin a. + x-. cos a,
*г 2г-1 г 2г ι
х2р+1 ' "x2p+j ■
x2p+t?+Jc и x2p-h7+fc '
the Index г varying from 1 to p, j from Ί to 9, and к from 1 to
n-2p-q. One of the Integers p,q,n-2p-q (or even two of them) could be zero.
The displacements proper from n-1 dimensional spherical space correspond
to orthogonal transformations with determinant Ί, I.e. which contain an even
number of roots λ«-1; the displacements accompanied by a symmetry correspond
to an odd number of such roots.
155. With this established, every holonomy group of a normal, locally
spherical space must only contain operations that do not fix any point. This
amounts to saying that the characteristic equation must not admit the root
λ- 1.
In fact, every point obtained by annulling the co-ordinates relative to
the roots different from 1 will be fixed by the operation In question. The
converse Is also true. Therefore we shall always have π - 2p + <?.
Let us firstly assume that the space has an even number of dimensions, I.e.
η га odd. The root λ-l never occurs; the root λ--1 will occur an even
number of times, consequently all of the non-identity operations of the holonomy
group are dieplaoemente accompanied by a symmetry. If S Is one of these
2
operations, the operations S , which represent a displacement proper, must
reduce to the Identity. This means that the squares e of the Imaginary
roots must all equal 1, which Is Impossible. Therefore all the roots of
the characteristic equation are equal to -1, and the only non-Identity
operation of the holonomy group Is
x\ я -х£ (г ■ l,...,n).
If, therefore, the considered space Is not spherical, we can obtain It by
regarding two points that are antipodal to each other as being Identified
with two points of spherical space. We then obtain an elliptic apace. The
proof even shows that this space Is non-orlentable. We have the following
theorem, of which the first part Is due to Killing:
Theorem. Every even-dimensionalt normal, locally epherioal apaoe is identical
to either ephavioal epaoe or to elliptio epaae. The first ie ori*ntabtet the
second ie not.

168
Bon-Euclidean Geometrice
In particular the elliptic plane (or the projective plane which Is
topological^ Identical) Is not orlentable.
Let ив now аавите that the space ie odd-dimensional, I.e. η Is even.
The canonical form of the orthogonal transformations shows that If the root
λ- 1 does not occur, then the root λ --1 occurs an even number of tines;
consequently the determinant of the coefficients Is equal to 1. The holonomy
group only contains the displacements proper. Whence the following theorem of
Killing:
Theorem. Every odd-dimensional normal, locally epherioal apace ie orientable.
In particular, odd-dimensional elliptic space Is orlentable. It
corresponds to the holonomy group formed by the operation χ', --x. and the Identity
operation.
IX. THREE DIMENSIONAL RIEHANNIAN SPACES SATISFYING THE AXIOM OF THE PLANE
156. We are now going to show that the only three dimensional Rlemannlan
spaces satisfying the axiom of the plane are locally Euclidean, locally
elliptic, or locally hyperbolic spaces.
Every sufficiently small region of such a space admits, as we have seen
(No. 114), a geodesic representation on ordinary space. Let us make this
representation. The Isotropic directions emanating from a point N give rise to a
second order (Imaginary) cone with vertex at N; we shall denote this cone by
(Гц). Let us consider any two points А,В (belonging to the region of
ordinary space on which the representation Is made), as well as the straight line
joining them. We have seen (No. 103) that any two planes containing AB
Intersect at A and at В with the same angle (defined by the Rlemannlan metric,
I.e., by the Isotropic cones with vertices at A and B). The Involution
defined by any pair of rectangular planes passing through AB Is therefore the
same at A and B. Therefore the double planes of this Involution are also
the same. Consequently, the tangent planes to the cone (Г.), that are taken
through AB. are also tangential to the cone (Г.): In other words, any too
isotropic сопев admit fcj© carman tangent planes*
From now onwards we could either take a geometric or analytic view of
things. Let us take the former.
Let А, В, С be any three points not on a straight line. Let us consider
the two tangent planes. P., Рд, common to the two cones (Γβ) and (Г.);
the two tangent planes ΡΒ·Ρή common to the two cones (Γ-.) and (Г.); the
two tangent planes Ρ-,ΡΙ common to the two cones (Γ.) and (Γβ); let us
finally consider a tangent plane Π. to the cone (Γ.) and different from
P., Pi, P., Pi, and likewise, two similar planes IL and IL.
The nine planes that have been defined are tangential to the same quadrlc
(Q). We are going to show that the ieotropia acme vlth vertex A ie
earibed by this quadrio*

fkm-Eualidean Geometries
169
Effectively, the cone with vertex A Is circumscribed by (Q), and the
cone (Гд) has five common tangent planes, I.e. Ρβ, Pj, Pc> P£, Пд; therefore
they are Identified. Likewise, we could show that the Isotropic cones with
vertices В and С are circumscribed by (Q).
Now let N be any point. The cone with vertex N, circumscribed by
(Q), and the cone (Гц) have six common tangent planes In common. I.e. the
two tangent planes common to (Гц) and (Г.), the two common tangent planes
to (Гц) and (Г_), and the two common tangent planes to (Гц) and (Г.).
Consequently, the cone (Гц) Is circumscribed by the quadrlc (Q).
The result of this Is that the different Isotropic cones are the cones
circumscribed by (Q). This quadrlc could degenerate Into a conic (necessarily
Imaginary); we could easily show that It could not degenerate Into a system of
two points. If It does not degenerate It Is either Imaginary, real and
unruled, for If It Is real and ruled, there would pass through every point N
real. Isotropic directions. Besides, If It Is real and un-ruled, the points
N of ordinary space that represent the points of Rlemannlan space are
necessarily Inside the quadrlc.
In the three possible cases, we could define a Euclidean, elliptic or
hyperbolic matrlc (and even an Infinity of them) for which the quadrlc (Q)
plays the part of the absolute. The Euclidean metric corresponds to the case
where (Q) reduces to a conic, whilst a holography could always transform
Into an umbilical. In each case, the (Euclidean or non-Euclidean) metric
depends on a unit of length that may be chosen arbitrarily. In the metrics
defined by the quadrlc (Q), the angle between the two directions emanating
from the вате point Η ie the вате aa in the given Riemonnian metrict since
the cone of Isotropic directions Is the sama.
Definitively, the required Riemonnian враоев enjoy the property of
admitting a geodesic and conformal repreaentation on a Euclidean apace, or on a
non-Euolidean, elliptic or hyperbolic apace,
157. It Is now easy to see that if too Riemannian epaoea together admit
a geodesic and confoimal repreaentation on each other, then the line elements
of these врасвв only differ by a conetant factor. The transport by parallel-
Ism of one vector χ of origin Μ to a point M" Inflnlteslmally near gives
two vectors x* and x' with the валю direatlon In the two spaces.
Effectively, using Severl's theorem, we make this transport by tracing the geodesic
HM", constructing the geodesic surface at H, tangential to НИ' at H,
and taking to H' a tangent vector to the geodesic surface such that this
vector makes the same angle with HM' produced as χ at N does with НИ'.
The construction gives the same final direction In the two spaces.
We are now going to show that the two vectors χ' and χ' not only have
the same direction at Μ', but are actually Identical.

170
Non-Euclidean Geometries
Let us denote then by UM and (иг + A1) the co-ordinates of the
points Η and H', and the components of the vector χ by Хг. Those of
χ" and of χ' are respectively
and
X1 - X* Г^ duh
X£ - X* rjh duh
where Г^ and ГГ^ are the Chrlstoffel symbols of the two spaces.
We must determine the same components for x' and xr to within a
factor 1 + ω, 1nf1n1tes1ma11y near to 1. we then have
**<4 - 4> л*.а* .
The factor ω Is Independent of the vector x, otherwise It would be equal to
η nomographic functions of the variables Хг with distinct denominators,
which Is absurd. The Immediate result Is the equality of the coefficients
r!, and rj. once the upper Index г differs from the lower Indices к and
h* We have furthermore for all it
"- <i< - γ«>λϊ :
this Is only possible If the quantity ω Is zero, with 7J, - Г., .
As the Chrlstoffel symbols are the same for both spaces, the lou of
parallel transport is equally the earn in the two spaces.
This being the case, and since the representation of the two spaces, one
2 2 2
upon the other, Is conformal, we will have de^ - к dst , where к Is a finite
factor (a function of и ,...,u"). Let us start from a fixed point A; a
vector χ originating from A Is measured In the two spaces by two numbers
whose ratio Is equal to the numerical value к of к at the point A. Let
us transport this vector χ by parallelism from A to Η along an arbitrary
but fixed path; It will give a vector x' which, In each of the two spaces,
has the same length as the vector x; consequently, the ratio of the numbers
measuring this vector In the two spaces Is again equal to ft . We therefore
have к ■ к » const. This Is what we wanted to prove.
The Μne element of every Rlemannlan space satisfying the axiom of the
plane only therefore differs by a constant factor from a Euclidean or non-
Euclidean (elliptic or hyperbolic) line element; It Is consequently either
Euclidean or non-Euclidean Itself, and we arrive at the following theorem-.
Theorem. Every Riemannian spaas satisfying the axiom of the plane ie either
looally Euclidean, locally elliptic with oonetant curvature.

Chapter VII
RIEMANNIAN CURVATURE
I. THE DISPLACEMENT ASSOCIATED WITH A CYCLE
158. We are now going to look at the most general Rlemannlan spaces In
a manner that clearly distinguishes them from Euclidean space.
We have seen (In No. 92) that for a given arc aob It was assumed that
the corresponding natural frame (R) was attached to each point and It was
possible to develop this arc on Euclidean space (e.g. on the Euclidean tangent
space at a), as It was for all corresponding frames. If the space In
question Is not locally Euclidean and If the two points a and b are not too far
removed from each other, then the development of another arc ao'b* originating
from the same point a and ending at the same point bt will give two
different results If It Is taken from the same Initial position A (with frame R.)
In Euclidean space, with the point b and Its frame occupying. In one case,
the positions В and (R«) respectively, and, In the other case, B' and
(R') respectively.
We can express the above result In the following way. Let us assume that
1η going from the Initial position B' and (Rg), In Euclidean space, the
closed contour or oycle ba'acb Is developed. After describing the cycle,
the final position В and R& for the original point b of the cycle and
Its corresponding frame are obtained. In order to recover the Initial
position. It will be necessary, in Euclidean epaoe, to produce a certain
displacement which takes В to B* and (R„) to (R'); this displacement Is said
to be associated with the given cycle. It very much depends on the Initial
position given for B* and (RJJ; but It could be said that It Is well
173

174
Riemannian Curvature
defined with respect to the frame R'. More directly, we could say that It
Is a well-defined displacement In the Euclidean tangent space at B'.
We add the following two remarks to help clarify matters.
Remark 1. If in Riemannian space, the body of vectors attached to b is
transported by parallelism along the length of ths cycle bo*aab, then it
experiences ok arrival an identical rotation to that which would have resulted
from the displacement associated with the cycle produced in the Euclidean
tangent space at b.
r+ -*■
Remark 2. If the vector integral I DX, where X denotes a given vector
field in Riemannian space (e.g. the vector field e,)a ie extended in Euolid-
ean space to the contour B'C'ACBj, we obtain the geometrical variation
undergone by the vector field attached to the point b, until the i nveree
displacement associated with the cycle bo*acb is applied.
159. It would be very difficult to make a direct study of the displacement
associated with an arbitrary cycle starting from a given point and returning to
It. We are going to restrict our attention to an infinitesimal cycle, and
assume this cycle has a particular form.
To this extent, let us consider two distinct systems of differentiation
that will be denoted respectively by the symbol ι d and *. The quantities
du1 may be regarded as the products of a constant, Infinitesimal parameter,
and fixed functions £г(и ,...,un) (which could have remained Indeterminate):
du1 - ai<(J un) .
Similarly, we shall have
6u£ = впг(|Л...,ия) .
Now let m be any point In Riemannian space with co-ordinates (uM.m, the
point with co-ordinates (u + du ), and я. the point with co-ordinates
(иг + 6иг). The vectors mm,, тт~ define elementary displacements d and
6 respectively. Let us now apply the elementary displacement to the point
тг·. we obtain a point π. with co-ordinates и + du + Aiu1 + du ) ■
d> + du' + fiu. + δώΛ Similarly on w-, we apply d, and we obtain a point
ml with co-ordinates
иг' + 6u£ + d{ul + бц*) - и1 + 6u£ + du + dbt .
It may be seen that the point m\ will coincide with ни If we nave
dtu1 - 6аиг ;
In other words, if the two differentiate are interchangeable. Thus If / Is

Riemannlan Curvature
175
an arbitrary function of the co-ordinates» we know that
d6f - 6df .
We shall assume that both differentiations are Interchangeable. Then every
point m of the space Is one of the vertices of an elementary ayote wn, т*
m2 that we shall assume follows the order Indicated by the preceding notation
(the sense of the rotation being that which takes the direction of the
displacement d Into that of £).
These are the elementary cycles that we are going to consider. In the
particular case where we have
(ξ1* ■ 1 the other ζ being zero
η* ■ 1 the other η being zero ,
the points M,, H?, H. have the same co-ordinates as π, except that for
the point m,» the co-ordinate U2* Is augmented by a, for m-· the
co-ordinate U* Is augmented by 8, and from M-, the two co-ordinates
\f and \f are augmented by α and β respectively.
160. When Euclidean space Is developed on the side лгт, of the cycle,
by going from a certain position (Μ,β.)· then the point Η and vectors β.
of the frame to which they are attached undergo elementary geometrical
variations given by the relations
dM я du e.
de£= ω* {d)ek
(1)
Similarly the development of nn. produces the variations
6H - iu\ (2)
6a£ - uk (6)efc .
Let us now develop the contour rm^my It Is necessary to apply the
operation 6 defined by (2) on going from M, and the frame R,. But the
relations In (2) give the operation δ at the point N, whereas It must be
carried out by replacing the co-ordinates и by u + du everywhere. In
other words, the point M3 and the vectors (ȣ)3 are deduced from the point
R, and the vectors («.)« by applying the operation d to them. We
therefore have In Euclidean space,

176
Riemannian Curvature
н3 = μ + &h + d(n + ем)
(е.), - бе. + в. + d{e. + δβ.)
г'З г г г г'
In contrast, the development of the contour MM* M. will give
M3 - Μ + eM + <5(M + αΛ\)
(вг.)3 = в< + dei + δ(β{ + de{) .
Consequently, 1n the displacement associated with the cycle the
point M3 and the vectors (03 undergo displacements
VM^ = M3 - M3 » 5dH - <ίδΜ
ν(β<)3- (*-)3- (^)3a №. -dfe. .
In practice the above quantities only express the principal part of the
displacements 1n question, being the principal part which 1s of the order aft.
I.e. the order of the area bounded by the cycle. These displacements are,
moreover, relative to the frame (R) attached to the point M; but to the
degree of approximation 1n question, 1t 1s possible to regard the e. 1η the
last set of relations, as vectors of the frame (Rl) attached to ML. The
calculation gives,
d6H - d6M a (6du - d6u W + [du 6e. - 6u de.)
= [du*u£ (δ) - δΛ£ d] eh = (Г^- r£r) du* 6u3 eh ■ 0
6dei-d6ei = [δω* (d) - diu* (6)]sfc + [ω£ (d)u£ (δ) - ω* (δ) cuh(d)]eh
3 ί^ (6,d)efc
by putting
flj(6,rf) - δω* (Л - du* (δ) - [u>J (δ)ω* (d) - ω* (d)** (δ)] .
By using the condensed notation already Introduced (see No. 45), we can see
that the alternating bilinear form 1s equal to
Я* - du* - [ω* и?]* ·* (3)
ъ t t η

Riemannian Curvature
177
Finally, we obtain for the displacement associated with the elementary cycle
1n question,
/ VH - 0
Uei - Ω* (6,d)ek
It can be seen that this displacement leaves the origin of the oyole fixed
and it ooneequently reduces to a rotation about this point.
161. There Is an alternative means of calculation. Let us attach
(Ideally) to each point m of R1emann1an space a point ρ defined by Its
Cartesian co-ordinates (хг) relative to the natural frame (R). In the
development of Euclidean space, this point ρ will naturally take the position
Ρ defined by Its Cartesian co-ordinates. In the development of the side ww,
of the preceding cycle 1n question, the point Ρ undergoes an elementary
absolute displacement whose contravarlant components are (see No. 44)
Dx1" = ** ♦ A* ♦ Λ£ (d) .
Similarly, 1n the development of mw., 1t undergoes a geometric displacement
with components
Δχ* ■ δχι" + fiu* + Aj (δ) .
The development of the contour πη.π. then gives a point P. which has
co-ordinates, with respect to the frame (R) of the point H,
xx + 0хг + Δτι + DAr*
The development of the contour tmjn~ gives, 1n contrast, a point PI with
co-ordinates
xi + Δτ* + Dxi + ΔΟλ1
The displacement undergone by the point P. 1n the displacement associated
with the cycle thus has contravarlant components
V*1 - ΔΟζ1 - DAr1
The calculation gives
Vxi ■ 6D*£ + θΛ£(β) - dtJ- - A*fcw£W) .
Now

178 Riemannian Curvature
6Qb* + θΛ£(β) - 6dJ + 6ώί1" + fa*«*U) + *fc 6o)jJ(ii)
+ αΛ{(β) + *A£(6J + Λ*(<ί)ω£(*) .
This 1nmed1ate1y results 1n
Vxl - ** {«SujU) - *£(6) - [uj(6)(^(d) - ^Ц(б)]}
+ [ώΛ»£(ό) - βΑ£(ί)]
The last sum 1s zero and we obtain
V** - ujf(u,d\xk · (5)
The fact that the term Independent of il 1s zero on the right hand side
Indicates that the elementary displacement associated with the cycle 1s a rotation
about the origin of the cycle.
The quantitiee il (6 ,<j) are the mixed oortyonente of the biveator which
repreeente thie rotation (see No. 20). The cova riant components of this Μ
vector are
пу(б.*) - itte£(6.*) .
Ые are certain in advance of the relatione
il.. + il.. - 0
ij ji
which, Moreover, can always be proved by calculation.
II. THE RIEMANN-CHRISTOFFEL TENSOR
162. Let us consider how the cycle Involves the components of the
rotation with which 1t 1s associated. The Я? are alternating bilinear foms of
к
the du ; 1n other words, we have
0i(d.6) - tfj" (*ρβ/ · du'bf) - УЙМ ρ
re
ire
with skew syeeetrlc coefficients 9?. with respect to the Indices r and β;
ti4e та
they are then linear with respect to the (contravariant) components ρ of
the Ыvector represented by the parallelogram which determine the cycle. If
then cP. are the mixed components of the Μ vector which represents the
rotation associated with the cycle, we have the formula
4ш vi «■ p™ ■ -wi» pre · (6)
Thie formula ie general, 1n the sense that 1t Is Independent of the particular

Rimannian Curvature
179
form of the contour of the cycle, provided 1t encloses an element of the
surface equivalent to a Mvector with components p™. We will accept this
result without proof. (In the neantlrae see Nos. 190 and 192). The quantities
Fr. define what 1s known as the Riemam Chrietoffel tanaor.
Calculating these quantities does not pose any problems. He obtain
"L-9S-i- [rf ri.-1* it.] .
β Ρ
163. Let us undertake to calculate the сover1 ant components of the Μ
vector which represents the rotation associated with a given cycle. On taking
Into account (3), we have by definition,
*ij - [илЛ3' [ИЛ/3 ;
The two different expressions obtained are equivalent, for we have
The relationship 1n (8) shows quite clearly that Interchanging ΐ and j1
changes the sign of Ω.,; the equation
*« " ttirf + "rt
yields, by noting that δ^{, ■ &fey.
0 ■ du.. * dw..
The actual calculations of the coefficients R.._ of Ω.. can be produced
without difficulty from the equations 1n (8). We have
Rw™ ■ ^ - ^ ♦ J* < W*. - W,*.)
Now

180
Riemannian Curvature
bTiJe _ Hie r] _ 1 / Э gjj , Э fra Э He \
Эй" ЭиГ 2 ^ эЛи* Эи*ЭиГ oJou* '
"tfi-, oiirj] _ 1 / *Z4d, ***&_ J^ir\
Эк* Эив 2 V эЛи' Ъи1Ъив Ъ^Ъи* '
Thus
Л Л Л Л
R - 1 / ЪЧг , Э>а »*i« Э gjV \ (9)
^~ * ^ Э^Эи* э7»7 э7»7 " аигаив '
* д01 lUr k][j* h] - Не к]Цг Ш .
The relationship obtained reveals the Important symmetric properties of the
Riemannian eynbote R., . We also have the relations
R.. - -R.. - -R.. , (10)
ijre jire ver
R.. - R .. , (Π)
ijWi i&ftj iJyfc
Furthermore, we can show that-relations (10) and (12) Imply (11).
III. RIEMANNIAN CURVATURE IN TWO DIMENSIONAL SPACE
164. In the two-dimensional case, tha displacement associated with an
elementary cycle reduces to a rotation about the origin Η of the cycle. If
do 1s the area bounded by the cycle, and К do the angle of this rotation,
oalculated in the positive aenee of the oourae of the ay ale 3 then the aoeffi-
aient К 1s called the Riemannian curvature of the epaoe at the point H.
12 12 12
If a ■ -ИЛ Ρ 1* the contrevarUnt component of the Mvector
representing the rotation, we have
К <fc - 7ξ α12 - -R]2 ή ρ12 - -R]2 da ;
consequently
к ■ -R ■ - — R
* K12 g K1212
One could well benefit by referring to The Theory of Surfaaee by G. Darboux
(Tome III livre VI).

Riematmian Curvature
181
Let us take as an example a surface 1n an ordinary space taken to be a
two-dimensional R1emann1an space. To have R1enenn1an curvature at a point A,
we can relate the surface with respect to three rectangular co-ordinate axes
Ar, Ар, Ал having this point as the origin, the ζ axis being normal to the
surface. By taking co-ordinates χ and y* and 1n tha usual way denoting
by Ρι<7ι><,βι£ι the partial derivatives of the first two orders of the
function д of χ and у which defines the surface, we obtain by calculation
tha form
where tha sunmtlon Index г takes values Ί and 2. Now the sun on the
right hand side has all Its coefficients zero at the point A, since dx +
2
dy 1s an osculating Euclidean line element to that of the surface at A.
On the other hand, we have
ds2 » (1 + p2)dx2 + Zpqdxdy + (1 + q2)dy2
ω12 * Γ121 Л * Γ122 * " [ΊΊ 2]d& * [12 2]dy
я q(r dx * в dy) * q dp
Consequently
du>12 - [dqdp] - [(edx * tdy)(rdx * edy)) - (β - rt) [dxdy]
We then have at A
К ■ -R1212 ■ rt - в2
But 1f we had taken as axes χ and y% the principal tangents at A, we
have, at this point
κ1 κ2
on denoting the two principal curvatures by J- and J- . We then have
Kl K2
K"4 ;
the Riemannian curvature of a surface at a point is equal to ita total
curvature, being a product of Its two principal curvatures. It was Gauss, as we

182
Riemannum Curvature
know, who first showed that the quantity 5Λ- only depends on the line
element of the surface. 2
165. We now focus our attention on a particular Instance concerning the
displacement associated with a cycle. Let us take a sphere of radius R and
a small circle with pole Ρ (F1g. 26). Let us denote by α the half angle
at the vertex of the cone, having as Its vertex the centre of the sphere and
the small circle as Its base. The development of the circle on a plane 1s
obtained by unfolding the developable surface circumscribing the sphere along
the circle; this development 1s a cone. Let us attach to each point Η of
the circle two rectangular axes, Kr tangent to the circle 1n the sense of
the chosen course and N# tangent to the meridian passing through N. The
development will give an arc of tha circle with centre S and radius R tan α
(see No. 101), the length of this arc being equal to the circumference of the
small circle of the sphere, I.e. 2irRS1na. If then φ denotes the angle
A'SA, we have
R tan a(2n - Φ) ■ 2irR S1n a
where
Φ ■ 2π(1 - Cos a)
On the other hand, the region on the sphere bounded by the circle 1s
S ■ //R' S1n*d6o> ■ 2nRb/ Slnfd*- 2irR2(l - Cos a) ;
//r2 S1n*d6o> ■ 2nR2/ S1n
We then have
Φ
S
7
It can be seen that the rotation taking the frame x'Ap' to be parallel to
xby 1s precisely the rotation of the angel φ taken in the веплв of the
ооитве of the cyole. This rotation 1s thus equal to the product of the area
S bounded by the cycle and the curvature —к of tha sphere.
R *
A translation taking A' to A, and whose magnitude 1s 2R tanaSIn *
1s combined with this rotation. Tt is поп-гяго; but 1f the cycle 1s
Infinitesimal, this translation 1s Infinitesimal 1n comparison with the rotation, tha
3
principal part nRa vanishing when we take the Infinitesimal principal area
bounded by the cycle.

Иглтстпгап Curvature
ТЧ- »;.
166. In the case of an arbitrary surface, a simple geometrical
Interpretation may be given-, this relates to a well-known theorem of Gauss concerning
the sum of the angles of a geodesic triangle.
Let us consider the cycle formed by a very small geodesic triangle on a
surface. If 1t 1s developed on the tangent plane at one of Its vertices A,
the rectilinear triangle ABCA' 1s obtained, where the point A' 1s
Identified with A to the approximation 1n question.
Г1|. »■
с
л

184
Riemarmian Curvature
The side AB» through which the developnent commenced, 1s tangent to the
side ah of the geodesic triangle. As for the direction of the side ασ, 1n
order for its determination it Is necessary to turn the direction of AC
through the angle Ыо. Now 1n this development, the angles Ё and £ have
been preserved; the angle ВАС of the rectilinear triangle 1s thus equal to
ir-B-C. We must find the angle Д of the geodesic triangle by turning AC
through angle Ыо\ consequently, we have
π-§-£ + Κίίσ - A
whence
А + В + С-тг-fcfo . (13)
This 1s Suass's theorem.
In the particular case of the sphere radius R, the left hand side of
(13) (I.e. the spherical excess) 1s effectively the area of the spherical tr1-
2
angle divided by R ; this follows from a theorem of Albert Glrard.
The R1emann1an curvature at a point of a surface (or any two dimensional
R1emann1an space) could be defined by the Hm1t of the ratio £ " * ·
1n which A,B,C are the angles of an Infinitesimal geodesic triangle with
area <fo, having the point A as a vertex.
167. More generally, any Infinitesimal cycle could be considered, 1n Its
development, the tangents 1n two neighbouring points H,H' of the points mjn*
of the cycle, are at an angle to each other, equal to the angle of the geodeeia
da
contact of the cycle, I.e. — where ρ denotes the radius of geodesic
curvature (see No. 100). 9
The direction of the tangent at the Initial point A of the developing
/da
— ; the angle through which
Я
1t must subseuqently turn to obtain the angle of rotation 2π 1s thus equal
to fefo, and we obtain the formula also due to Gauss,
/
P9
Under this guise, 1t could be extended to any cycle and so gives
>~ft-fh ■
The theorem relating to the geodesic triangle 1s only a special case. In the
case of a geodesic polygon, the resulting formula 1s, on denoting by я the
sum of the anales of the polygon and η the number of Its sides,
β - (η-2)π -// fcfe . (14)
■Я

Rtemarmian Curvature
185
168. The preceding formula leads to some Interesting results 1f 1t 1s
applied to a closed R1emann1an space with an everywhere regular metric. If
this space 1s decomposed Into geodesic polygons, and F 1s the number of them,
S the number of vertices and A the number of their sides, then the sun of
the left hand side 1n equations (14) gives for ewry polygon.
2tt(F + S-A)
we then have
F + S - A - ^ / / tda
*//■
the Integral being extended to all of the space.
We know that the Integer F + S-A only depends on the topological
properties of the space. If the space 1s homeomorphlc to the surface of the sphere,
this Integer 1s 2» and we have the formula
//
fcfo - 4π
which shows that the least value of the Riemarmian curvature of the apace is
4tt
-ξ-, where S denotes the total surface of this apace. It 1s therefore
Impossible to define on a manifold, homeomorphlc to the sphere, a metric adnlttlng
an everywhere zero or negative R1emann1an curvature.
If the space 1s homeomorphlc to the elliptic (or projective) plane the
least value of the R1emann1an curvature 1s equal to ^ . If the space 1s
homeomorphlc to the torus, this least value 1s zero. We have 1n fact seen that
the torus may be endowed with an everywhere Euclidean metric.
All of these results are moreover evident on the closed surfaces of
ordinary space, where the quantity \\Ыз% due to a theorem of Gauss, denotes
the oriented area of the spherical representation of the surface (aurvatura
integra).
IV. RIEMANNIAN CURVATURE IN 3-DIHENSIONAL SPACE
169. To study R1emann1an curvature at a point A 1n three-dimensional
space, we shall assume, without loss of generality, that the frame attached
to A 1s rectangular. The Riemann Christoffel tensor Involves six quantities
that for the sake of brevity we shall denote as follows. We establish

186
Riemmnian Curvature
7323
l3131
Ί212
R ж R
K3112 K1231
R1223 B R2312
R в R
K2331 K3123
- К
11
-«22
k33
31
(15)
- К
12
Every elementary cycle originating from A could be defined by the suppV
tary vector of the Ыvector bounded by the cycle. If α,β,γ are the direction
cosines of the normal to the plane element containing the cycle, and 1f da
denotes the area bounded by the cycle, then the quantities
ode, &do, у do
define the Ыvector bounded by the cycle.
The rotation associated with the cycle could be similarly represented by
a vector taken on the axis of rotation; we shall denote Its components by
pdo, qda, yda
We have
pdo ■ -(R23230iio + κ?331βίίσ + Κ2312Ύ*^ * ^Κ11α + Κ12β + Κ13γ'ίίσ ■
Consequently we obtain
Ρ*Κ11α+Κ12Β + Κ1*γ
q * K^a + K^B + KgjY
r - K31a + K32B + Κ33γ
(16)
We shall in the future call the nomal projection of the vector [p3q,r),
representing the rotation aloee to unity of the aurfaoe as во dated with the plane
element in question, the Riemoowian curvature К of the epaae at A in the
direation of that same plane element. He then have
К ■ pa + q$ + ry
' К11°2 + *22β2 + Κ33γ2 + 2^3βγ + ΖΚ3ΐΎα + 2Κ12αβ
(17)
The right hand side of this expression 1s a quadratic for* that we shall denote
by Φ(α,β,γ).

Biemannian Curvature
187
170. It 1s Important to note that knowing the R1emann1an curvature at
A of the space 1n every planar direction Involves knowing for certain the
R1emann-Chr1stoffel tensor which 1s defined precisely by the six coefficients
of the form Ф. The law of variation of the R1emann1an curvature 1n a variable
planar direction could be formulated geometrically 1n a simple way. Let us
consider, 1n the Euclidean tangent space at A, the quadrlc centred at A
with equation
♦U.*.a) ■ 1
He refer to 1t as the R1emann1an quadria indioatrix.
The R1emann1an curvature 1n a given planar direction 1s obtained by taking
the normal at A 1 η this direction and one of the points Ρ where 1t meets the
quadrlc 1nd1catr1x. He then have
if - !
K "i? ■
Effectively, with the co-ordinates of the point Ρ given by αρ. $ρ, γρ. we
have
1 - Φ(αρ,$ρ,γρ) - ρ2*(α,Β,γ) - Kp2
whence
к 7 ■
The quadrlc 1nd1catr1x, 1n fact, could be an Imaginary ellipsoid (1n which
the spatial curvature 1s positive 1n every direction), a real ellipsoid (with
negative curvature 1n every direction), a hyperbolold. an elliptic or
hyperbolic cylinder; or finally a system of two parallel planes. The quadrlc
disappears if the space 1s Euclidean.
The space 1s said to be isotropic at the point A 1f the quadrlc 1nd1ca-
trlx 1s a sphere: we also say that the spaces at A have constant (locally)
R1emann1an curvature.
If we now return to the relations 1n (16) giving the rotation close to
unity associated with a cycle, we can easily see that the arte of this rotation
ie perpendicular to the conjugate diametral plane of the normal· to the ay ale
with respect to the quadria indioatrix. The rotation 1s made about the normal
to the cycle 1n this case, and only 1n this case, where this normal 1s one of
the axes of the quadrlc 1nd1catr1x; the directions of the axee of the quadria
indioatrix are catted the Riaoi principal directions of the erpaoe at the
point A. They are Indeterminate 1f the space 1s Isotropic at A.
171. He are now 1n a position to account for the fact that totally
geodesic surfaces only present themselves exceptionally 1n any R1emann1an space.

188
Riemannizm Curvature
In fact» 1f a surface 1s totally geodesic, the normal to the surface
transported parallel to Itself along a cycle traced on the surface, staying normal
to the surface, must return to its initial position. Consequently the rotation
associated with any elementary cycle traced on the surface must be made about
the normal to the cycle. The totally geodesic eurfaae must then, at each of
ite points, be normal to one of the principal directions at this point.
This theorem, due to R1cc1, Indicates that 1n general, there 1s no totally
geodesic surface. If the principal directions are distinct, there exists at
most three one-parameter families of totally geodesic surfaces. In this case
1t 1s necessary that each of the total differential equations expressing the
normal to the surface as a principal direction should be completely integrahle.
These conditions are not, however, sufficient. A closer examination reveals
Euclidean apace ae the only one admitting a triply~orthogonal system formed by
totally geodesic surfaces.
172. We can now extend to three-dimensional spaces the theorem of Gauss
concerning the sum of the angles of a geodesic triangle. Let us take a point
A 1n R1emann1an space and a plane element stemming from this point. Let us
take through A two very small geodesic arcs Ab and Ac tangent to this
plane element and joining the geodesic be. We develop the cycle Afce on the
Euclidean tangent space at A, and let ABC be the rectilinear triangle so
obtained.
Гц. *».
We take three rectangular axes attached to A, the axis Ax taken along
AB, the axis Ax being 1n the plane element 1n question, and the axis As
perpendicular to this plane elenent. Finally we shall represent the tangent
Ay to the geodesic Ac at A. A passage from Ac to Au 1s made by the
rotation {pdo, qdo, vda) associated with the cycle Abo. Conversely, we can
go from An to AC by the rotation (-pda,~pda>-rda). Let us consider the
angle made by Ax with the direction Au to which the rotation 1s applied.
This angle takes as Its Initial value A and as Its final value, π-B-C. Now
the rotation (~pdoM~qda,~rdo) could be decomposed Into a further three, one

Riemannian Curvature
189
taken on Au, one on AB and the third (-rda) on Ал; each of the first
two does not alter the angle made by Au with A*, the final one reduces 1t
to rda. We then have, with AC clearly 1n the plane xfy,
where
A - (w-B-C) - rda
A + В + C-TT ■ rda
Now r 1s precisely the spatial curvature Κ 1η the direction of the plane
element 1n question. We obtain the generalised formula of Gauss,
A+B+C-w-kto
173. There 1s a generalisation of the theorem from which the R1emann1an
curvature of a surface embedded 1n Euclidean space 1s equal to Its total
curvature.
Let us consider 1n any R1emann1an space, a surface S, and choose on
this surface an arbitrary system of co-ordinates utv. Every point Ρ of
the space, sufficiently near to S, could then be defined by the co-ordinates
utv of the point N of the surface obtained by dropping the normal geodesic
PH from P, and by the length i> of the arc MP. With this co-ordinate
system, the spatial line element takes the form
2 2 2 2
da ■ g-,*du + ΐρ-,^ώ^ + 9γ^° + ^°
The line element of the surface 1s deduced by putting ν = 0 everywhere:
2 2 2
de ■ g-t-tdu + Zg^dudv + g^d°
It can be seen quite clearly that at every point of the surface, the
quantities gtJ (i,j ■ 1,2) take the same numerical values, whether they are those
calculated from the spatial line element of the ambient space, or those from
the line element of the surface. In both cases, we have
11 12 22 ι
ff22 *12 *11 ffnff12-(?12)2 "
Similarly for the quantities Г., . and r!. , for which the Indices take the
values 1 or 2: the quantity Г., . - [ij3k)t only Involves the derivatives
of the coefficients #,,, g^ or д??* **ken with respect to u or υ; as
for the r!., they are deduced from г., . by means of the coefficients
11 12 !L 22
g . g and g .

190
Rternannian Cuwature
With this established, the Intrinsic R1emnn1an curvature of the surface
1s given by the form Ω_ calculated 1n the two dimensional R1enunn1an space
constituted by the surface; on denoting 1t bjr (a12) * ** "***
with the summation taken over the Indices к ■ 1,2.
The R1emann1an curvature of the ambient space, 1n the direction of the
tangent plane element to the surface, is given by the form
{Ώλ2]ο ' ^2 + Ιωϊ "2k] *
with the summation taken over к - 1,2,3, we then have
«Vo - (α12}β - Εω1 *23]
As a result of this relation, the difference between the two R1emann1an
curvatures at a point N on the surface only depends on the numerical values of
the coefficients of ω, and uu. at this point, and consequently dose not
change if the given metric is replaoed by a metric osculating at H. Let us
take as an example a Euclidean osculating metric; the difference К. - К
then reduces to the R1emann1an curvature of the surface, which 1s equal to the
product -K-K- of the principal curvatures. Now the principal curvatures are
12
the same 1n Rlemann1an space as 1n the osculating Euclidean space, consequently,
the general relationship 1s obtained
h - \ ■ jfc ■ (IB)
It expresses the fact that the intrinsic Riemannian curvature of a surface
at one of its points N ie equal to the Riemamian curvature of the ambient
space at N in the direction of the plane element tangent to the surface,
augmented by the total curvature (the product of principal curvatures) of the
surface at M.
In particular the Riemamian curvature of the epaoe. at a point N in the
direction of a given plane element ie equal to the intrinsic Riemannian
curvature at N of the geodesic surface at that pointy tangent to this plane
element.
The definition of R1emann1an curvature, as given by Rlemann, could apply
to the above considerations. Rlemann made an exact study of the geodesic
surface at N tangent to a given plane element and total curvature attached to
the line elament of this surface from a Gaussian point of view; this 1s known
as the curvature of the space at И 1n the direction of the given plane
element.

Riemannian Curvature
191
V. RIEMANNIAN CURVATURE IN A SPACE OF MORE THAN THREE DIMENSIONS. SPACES WITH
CONSTANT RIEMANNIAN CURVATURE
174. In a space of dimension n, the rotation associated with an
Infinitesimal cycle 1s represented by а Ыvector. The Riemannian curvature of
the space in the direction of the plane element of the cycle would be obtained
by taking the scalar product of the preceding bivector and the bivector bounded
by the cycle, and dividing the result by the square of the area bounded by the
cycle. It 1s easy to see that this definition coincides with that given
previously, for n-3. The scalar product of the bi vector (ado3&do,yda) bounded
by the cycle and the bivector {pdotqdotrdo) which represents the rotation
associated with the cycle, 1s
(pa + q& + ry) do
Uoc
Generally, by taking ptJ as the contravaHant components of the bi vector
bounded by the cycle, and p.. as the covaHant components of the associated
rotation, we have
a..
■ ч hm p
kh
and consequently
Kda2 -
or further still.
- * »v»* P^P
ijkh
ijjdi
ij kh
4p p.j
(19)
More generally, we may define at a point the mixed curvature (see
E. Bomp1an1) of two directions of oriented plane elements defined by the simple
blvectors ptJ and qtJ by the expression
* d ij kh
175. It 1s Important to note that knowing the Riemannian ourvature at
a point Μ of the epaoe3 in the different planar directions from M, involves
hunting the exact Riemann-Chrietoffel teneor at И. In other words, the
Identity
E. Bomp1an1, Spazi Riemanniani luoghi di varieta totabnente geodetiahe (Rend
C1rc. matern- Palermo, t.46, 1924).

192
Riemannian Curvature
assumed true for all simple bivectors ptJ, Involves term by tern equality for
every coefficient R..^ and ^■■fcA- It could 1n fact be said that 1f the
Riemannian curvature of the space at a point Η 1s zero 1n every planar
direction from M, then all of the components of the Rlemann ChHstoffel tensor
are zero at this point.
This theorem 1s far from evident, because if we assume the Identity (20)
as true, the variablee ptJ are not independent, the components of a Simple
bivector being constrained by distinct quadratic relations. He shall
Introduce Independent variables and define a Mvector by means of two arbitrary
vectors X1 and Y1; the Identity in (20) becomes
R..,, x
WxM - Б..,. xlWx*Y*
bjkh ijkh
valid for all varibles X1 and Y1. Be equating successively the coefficients
(Xt)2(W)2, (X^W and X*xVy* (where 1n each case, the written Indices
are all distinct), we obtain
К ■ ■ » t К · · ■ ■
R, .., + R.. ,, ■ IT.,., + Б.. ., (21)
tjtfc гкЦ ijvk ikij
Rijkh + Rkjih + RihkJ + Rkhij ' ^ijkh + \/th + RihkJ + 4hij ·
The relations derived 1n Mo. 163
at first show that the components R.... and R---k are the sane in the two
spaces. Moreover, on account of (21) and (11). we have
ij Юг i Hkj ijkh ihkj
where
Rijkh ' Rijkh я hhjk ' Rihjk
and by a cyclic permutation on the indices j\ fc, ft» we then have
КгВД " *ikhj я Rijkh ' *ijkh ' Rihjk " *ihjk '
The common value of these three differences 1s zero, since their sum 1s zero
by virtue of

Riemannian Curvature
193
hm + hkha + hhik" ° {12)
which the components Ж and R satisfy (No. 163).
We may note that the proof only Involves the properties of the \jkh
expressed by the equations (11) and (12), and
ijkh jtfch vQhk
176. He now restrict our attention to the case where the space 1s i&o-
tropic at the point in question, that 1s it admits the same Riemannian
curvature К in every planar direction emanating fron that point. The equation in
(19) then becomes
As the coefficients of the quadratic form in pv on the right hand side
satisfy the relations (10). (11), (12) exactly, we have
hjkh ' -«««ij-fc * HtPik^ or mre intu1t1ve1y» <22>
"да'*"-1"* ' {23)
This equation expresses the fact that the covariant components a., of the
rotation associated with an elementary cycle ρ * are of the form
av - К Ру (24)
Geometrically, the rotation te represented by a simple bivector situated in the
plane element of the oyale, and vhoee magnitude ie obtained by multiplying the
area bounded by the oyale by K; the rotation has the same sense of the cycle
if К 1s positive, and the opposite sense if К 1s negative. The number К
measures the Riemannian curvature of the space at the point Μ without having
to specify the direction in which it 1s taken.
177. The above property has a converse. If the rotation aeeooiated with
any elementary oyale attached to Μ ie represented by a simple biveotor
situated in the plane element of the oyale, then the epaoe ie isotropic at M.
The relations
ill.in !ϋ,„„
p12 p13 pij
which are, by hypothesis, true for every bivector, actually provide a set of
rational functions in the 2n independent variables χΊ,...,χ ; υ,,.,.,υ ,
ι η \ η
which allow us to define any simple соvariant bivector. These rational

194
Ri-emannian Curvature
functions reduced to their si ablest for» awst give the same Irreducible
quotient. Now the denominators X.Y. - X.Y. are evidently the first among then;
*■ j j t
consequently the common value of the functions in question is an integral
polynomial, by necessity of degree zero, I.e. a aonetant. We thus have a general
relation in (24), where К denotes a constant that 1s conveniently chosen.
178. One case 1s where the isotropy of the space at a given point N
1s known in advance; it 1s when the space has free mobility about the point M.
There 1s then an isometry leaving И fixed and transforming any plane element
from N Into some other plane element. This transforation manifestly
preserves the R1emann1an curvature (which only depends on the line element), and
the result of this 1s that the space must have the same R1emarm1an curvature
In every planar direction about M.
Tf the space satisfies the axiom of free mobility, then 1t 1s then
Isotropic at each of Its points, and moreover. Its Riemannian curvature 1s the
same at all points; It 1s said to have oonetant HemarvrCan curvature. Locally
elliptic or hyperbolic spaces must have this property, since they satisfy the
axiom of free mobility. This could be verified by a calculation in the
following way:
Let us consider the absolute quadrlc in a projective space and denote a
point by Μ (with squared scalar equal to yr . К being the curvature of the
space, in the sense of the word as given in Chapter VI). We also take the
points β,,.-.,β situated in the polar hyperplane of м with the respect
to the absolute quadrlc, these being analytically defined in the same way as
the basis vectors of the natural frame attached to M. We have proved (see
No. 146) the relations
IdM ■ du в.
л r *A. ♦ * {25)
in which the symbols <M and ie. Indicate the exact differentiate.
To calculate the соvariant components Q.,. of the rotation associated
with an elementary cycle, we firstly calculate άω...
ω.. ■ de * · β ·
consequently
(&£, - [de.-de^] - [a^^J + K[ff£i;du 'Q^du J . (26)
From this we can straight away deduce that
from which
%■' -ΚΙ»ΐ*** 'j***1

Riemannian Curvature
195
a.. ш Kp..
where the p., are the covenant components of the Μ vector bounded by the
cycle.
The space thus has constant Riemannian curvature K.
The above calculation leads to another Important result. The
iηtegrability conditions of the equations in (25), where К and «· are unknown
geometric functions in a projective plane (where an elliptic or hyperbolic metric
has been defined with given curvature K), are precisely the equations in
(26) that express the fact that the Riemannian curvature has a constant value
K. Consequently, the entire epaoe uhose Riemannian curvature га constant is
locally elliptic (if K> OJ, hyperbolic (if K<Q), or Euclidean (if К-О;.
The development of this space on spaces that are elliptic, hyperbolic or
Euclidean with curvature к is in fact possible, since the equations in (25) giving
this development are completely integrable.
179. We can now verify the theorem following which the axlon of the plane
1s only true for Isotropic spaces at each of their points. Let us assume, in
accordance with No. 115, that all (n-1) dimensional geodesic manifolds at a
given point A are totally geodesic. We consider one of these manifolds and
an elementary cycle steaming from A and traced across this manifold. The
bivector representing the rotation associated with the cycle could be decent-
posed Into simple Ыvectors situated in the (n-1) dimensional tangent space
at this manifold and a simple bivector situated in a plane element containing
the normal to the manifold. The rotations representing the first Ыvectors
leave Invariant the unit normal to the manifold; the result of this 1s that
it must be the same for the rotation represented by the last bivector, this
being zero. This being the case, if we take an elementary cycle attached to
A, the rotation to which it 1s associated will necessarily be represented by
a simple bivector situated in the plane element of the cycle; otherwise we
could find a normal vector to this plane element that 1s not invariant wider
thie rotation. The (n-1) geodesic manifold V , at A, normal to this vec-
n-i
tor would not then be totally geodesic.
With the rotation associated with an arbitrary cycle being represented
by a simple Ыvector situated 1л the plane element of the cycle, the space Is
Isotropic at A (No. 177). Tf then the axiom of the plane ie true in the
epaoe, then ЬЫ epaoe ie isotropic at sack of its points. He shall see in the
following Chapter (No. 199) that this property Involves the property of
constant Riemannian curvature.

196
Riemcmnian Curvature
VI. THE CONTRACTED CURVATURE TENSOR. PRINCIPAL DIRECTIONS.
180. The R1emann-Chr1stoffe1 tensor admits a contracted tensor
R.. - R^..
The symmetric property of the tensor can quite easily be shown by using the
relations in (1i)
The second contracted tensor R - R* - R*? 1s called the eoalar Hemannian
curvature of the space.
The set of directions stemming from a point and satisfying the relation
KjjAfdJ ■ 0
defines a second order cone attached to this point; we call it Rioai'e Cone.
The prinicpal directions of this cone are what R1cc1 calls the principal
directions of the space at ths point in question/
The principal directions are indeterminate If the space 1s Isotropic at
the point in question, but the converse 1s not true, in general. When such
circumstances present themselves, there exists in every case an Isotropy of
a second kind, larger than the one considered up to the present. The two 1so
troples are Identical for n-3; we shall return to this Idea later 1n
No. 200.
C. R1cc1, Direzioni e inoarianti prinaipali di una varieta qualunque (Att1 R.
IstU. Veneto t. 63, 1904, pp. 1233-1239).

Chapter VIII
THE BIANCHI IDENTITIES
I. EXTERIOR DIFFERENTIAL FORKS*
181. We refer to differential forms with exterior multiplication, or more
briefly, exterior differential forms, as those forms which occur under the sign
of summation in multiple Integrals; these obey certain rules of calculus that
we shall mention briefly.
For example, let us take in ordinary three-dimensional space, a double
Integral ι across a region of the surface:
I - I hdydz + Qdzdx + Ысау .
In the differential form
ш - Ρ dydz + Qdzdx + Rdxdy 3
the terns dydz, dzdx, dxdy are not absolutely Identical to ordinary products.
If the co-ordinates of a point on the surface of Integration are expressed
as a function of two parameters a, β, then a, 6 could be regarded as the
co-ordinates of a point in an auxiliary plane. The Integral ι would then be
reduced to an ordinary double Integral extending across a certain region of
this plane. To achieve this reduction, the symbols
dydz, dzdxt dxdy
are replaced by the quantities
fcH6' ifeft*»· №#** respective^.
Clearly then, dydz must not be confused with dzdy which 1s regarded
as being equal and opposite to dydz.
The above expressions can be represented as follows. We Introduce two
symbols of differentiation d-, and d~ and put
vis* * vHd6
where these two symbols are Interchangeable. With this notation we obtain
One may refer to E. Cartan, Lecons our lea invariants integraux (Paris,
Hermann, 1922).
197

198
Bumahi Identities
d}y d^M
d^ dLf
dmdx ■
d-,ζ d-,χ
ауЯ d~x
aedy я
dyX dyy
df d^
The quantities dydz, dsdx, dxdy may be regarded as well defined products,
but the multiplication 1s exterior (Grassmann) where the sign of the product
changes when the order of the factors 1s changed.
In a more general way, two arbitrary Interchangeable symbols of
differentiation, d, and dyf could be Introduced. If the results of the two
operations d, and dy are infinitesimally small, the surface of Integration
could be decomposed Into a network of curvilinear parallelograms, the vertices
of each of which would be
x, у, ж
χ + d-,χ, у + d-iy, ζ +■ d-*i, and
χ + dyx + dfdyXt у + dyy + d^dyy^ ζ + d-,ι + d^dyM
χ + df, у + d^, ж +■ d^z, respectively,
and the Integral ι would be the sum of the quantities
?(d}yd2z - dyzdg) + ЕЗЦ**^ - dyxdf) + R{dfdg - dydf)
extended across all the elementary parallelograms. In fact the quantities
dydz, dzdx, dxdy are taken to be the components of a simple bivector.
To avoid confusion, we shall make a convention: an exterior differential
product 1s put Inside brackets when such a product does not belong under the
sign of Integration (cf. Nos. 45, 147, 160, 163, 178).
182. The above discussion can be extended to multiple Integrals in any
number of dimensions leading to differential foras represented by sums of terms
such as
Mdxydx2 ■·. dxn]
The exterior product between brackets takes the place of a p-th order
determinant Involving ρ Interchangeable symbols of differentiation. Such a
product changes sign when two of Its factors are Interchanged (the coefficient
A naturally enough 1s not regarded from this point of view as a factor).
Having been given two exterior differential forms ω, and tL with
respective orders ρ and q, we shall define the extertor product [ω. ω*]
of the two forms as a (p + ?)-th order form obtained by taking in all

Bianohi Idtntitiea
199
possible ways, the exterior product of a tern of the first form and one of the
second with respect to the order in which the differentials are represented.
If, for example, we have
ω, ■ a*dx uL ■ b.. [ахгс1яР]
we shall obtain
CdJdxk] .
183. On the whole there 1s a series of Important formulae which permit
the transformation of a Multiple p-th order Integral extended to a closed
domain to a Multiple (p + l)-th Integral extended to a domain that actalts the
first one as Its boundary. The most simple one of these formulae is the one
given by Cauchy-Green,
Jp<b ♦ Q* - ff(jjl - f) ** . (l)
As a consequence of this we have Stoke's formula
(2)
♦»-©** ·
and then Ostrogradsky's formula
J hdydz + Qdzdx + ЪЫу 'ίίί(^ +ty + $) <ЫуаЯ ' (3)
For spaces greater than three dimensions there exist similar formulae.
The operation that brings «bout all of these could be presented under a
very simple guise. Firstly, let us consider the case of a simple Integral
ju(d) across a closed curve С Let S be a region on the surface (in n-
dimensional space) bounded by С We Introduce 1n S two Interchangeable
symbols of differentiation, d, and <*_> and divide S into a network of
corresponding Infinitesimal parallelograms. If m 1s the vertex of one of
these parallelograms (Fig. 30) and if m, and ли are ^е vertices derived
from the operations d. and <L, we then have

200 Bianohi Identities
гтз_
/ ω * шЦ ) + α'2 шЦ) ;
consequently the Integral /ω along the contour of the parallelogram 1s
equal to
u(d}) + CW2) + d} u(rf2)3 - [ω(^)+ ^2шЦ)3 - u(d2)
ж <^ ω(<*2) - rf2 ω(^)
PI* iu.
и-
The expression on the right hand side 1s known as the bilinear oovariant of the
expression ΰ (see No. 45). For example if Pdx 1s a tern fron ω, we have
d}{Pdf) - d2{?d}x) - d} Pd^e - <*2 9d}x ■ [dPdrJ
We then have Stoke's formula
/Par + Q4i/ + Rda - IIdPdx + dQ<& + dRdx , (2)
which extends to any number of variables.
On denoting by ω the differential fore under the sign Λ we shall
in turn denote by du the corresponding form under ff and call it the

Bianohi Identities
201
exterior differential of the form ω. We note that the exterior differential
form of ω ίβ zero if ω ie an exact differential.
To transform a double Integral Into a triple Integral, we now Introduce
Into the 3-dlmenslonal domain of Integration three Interchangeable symbols of
differentiation, thereby decomposing It Into a network of elementary parallel-
plpeds. We will show that the double Integral JJ ω taken across the surface
bounding one of these parallelplpeds Is equal to
d] ш(<*2,<*3) - <*2 b)(<fj,<f3) + <*3 ω(<ί] .rfg)
Let Kdxdy then be one of the terms of ω.
We can easily verify the Identity
+ <f-
d,A ίίοΑ rfJl
d-\X dgt djc
dyy d\g dy
+ d.
and consequently we have
fh'fff
dAdxdy
Hore generally If
ω c A. .(jr.(jr.
we have
//-■///л«*<*г///с
where cL· denotes the exterior differential of the form ω.
The procedure Of the exterior differentiation as Indicated for forms of
order 1 and 2 can be extended to forms of any order: the exterior differential
of the form
ω - A[(jri(jrp ··· dx ]
The notation Is that of E. KShler, Einfiihrung in die Theorie der System von
Differentiagleichungen [Hamburger Math. Einjteleohriftena 1934). It Is
preferable In several respects to the notation ω* used by E. Cartan and which
appeared In the first edition of this book.

202
Bianoki Identities
IS
d5 - [dMx-,dx0 ... dx ]
1 2 ρ
In all events we can make this a convention to be recognized for a while, at
least to the end of No. 185.
184. The Integrals which occur In the formulae of Green, Stokes and
Ostrogradsky.and generally In the relation*
Ι ω ■ I efH ,
(where ω denotes an exterior differential form of degree ρ and where the
right hand side Is extended to a (p+l)-d1mens1onal domain D In the space and
the left hand side to a closed p-d1mens1onal manifold V which bounds D) are
only meaningful If the domain D and the closed manifold V bounding It are
oriented. The domain could be oriented by the convention that a certain (p+1)-
hedron formed by the (p+1) Independent vectors β-,,β^....e^ attached to a
fixed point Is direct. The manifold V will be oriented In a coherent way,
by taking from each point И of V, a vector β' In the exterior of the
domain D and ρ tangent vectors β' β' ...β' at V such that the (p+1)-
hedron fromed by the (p+1) vectors ej,e' ,βΐ,.,.,β', Is direct. The orlenta-
tatlon of V is then obtained by the convention that the p-hedron formed by
the ρ vectors βί,...,β'. Is also direct.
1 ρ
It Is easily proved that Green's formula (and Stokes formula,which Itself
Is a generalisation) Is true under the above conventions; the same applies to
Ostrogradsky's formula. For example. In the case of Green's fomul a, If the
area of the xy plane over which there extends a double Integral Is oriented
anticlockwise, then the sense In which the Integral JPdx + ОД must be taken
about the contour, will be the same as that of an observer who moves around
It with the area to his left.
185. Exterior differentiation possesses above all some very
straightforward properties. Firstly, let ω be any exterior differential form, du its
exterior differential, and m a factor that Is a given function of the
variables. We have
а[тш) ■ mdu + [dnu] . (4)
Effectively, If an arbitrary term Is taken In ω of the type
*The relation [f[x)]b - [b f[x)dx Is a particular case of this general rela-
a J a.
tlon; f(x) could be regarded as a 0-degree form.

Bianohi Identities
203
Α[ώ] ... dxp] ,
the corresponding terms In /πω Is
mA[dr,dr2 ... dx] ,
whose exterior differential Is nanlfestly
midfidx-, ... dx ] + Α[Λπ tic, ... c£e] ;
addition of all similar terms proves the theorem.
A more general relationship Is the following. Let ω, and UL be two
exterior differential forms of respective orders ρ and q. Let us consider
the form [ω, ω?] of order [p + q). Let
Mdx. ... dx ] , B[^1 ... dy ]
be two arbitrary terms of which one Is In ω, and the other In oL.
Corresponding to them In [ω, ω«] Is the term
AB[dr, ... dx dy* ... dy ] ,
whose exterior differential Is
BtdAir, ... dx dy. ... dy ] + A[dBdx, ... dx dy* ... dy ]
The second term could be written as
(-if Α[<ώτΊ ... dr dudy} ... dy ] .
From this results the relation
d [ΰΊ ώ2] - №Ί i^] * (-1)^ [ώΊίίω2] (5)
which can be generalised for the product of any number of factors. This
generalises the product rule of ordinary differentiation.
We can now show that the operation of exterior differentiation, as defined
In the general case at the end of Ho. 183, Is covarlant for all changes of
variables In the sense that If a change of variables Is made from x. to #.,
and If by this change, the form Z[x9dx) is transformed to i>{y,dy)> the
exterior differential du (calculated by regarding the x. as Independent
variables) Is transformed to <$> (the y. being Independent variables).
Effectively the term kidx.dx- ·.. <*0> assumed to be expressed In terms of
the variables y., has its exterior differential equal to {dMx]dx2 ... daP],
In accordance with the product rule of differentiation, since the factors

204
Bianohi Identities
dx,,dx~t..*dx being exact differentials, have their exterior differentials
1 ' F
equal to zero (No. 183).
186. There Is an Important theогея due to H. Po1ncar6 on the successive
exterior differentials of any differential form: The second exterior
differential ie identically zero.
With respect to a linear form In ordinary space
ш - Pcic + Qdy + JUa ,
the theorem is evident. In effect, let us consider any closed surface S and
the Integral ffdu taken across this closed surface. We divide the surface
Into two parts S1 and S« by a closed curve С The Integrel ffdu taken
across S, Is equal to the Integral J ω taken along the curve С In a certain
sense; the integral //<£> taken across 5? is equal to the integral у ω
taken along С In the opposite eenee. The result Is that the Integral fjdu
across S Is zero, whatever the surface S. The exterior differential of
dw Is thus Identically zero.
The general analytic proof Is straightforward. Let
A[dr,(ir0 ... dx ]
1 Ζ ρ
be a tern of the given form ω; the corresponding tern of du Is
[dAar. ... dx ]
To make the exterior differentiation of this form, we nay regard ft as the
exterior product of (p + 1) factors, each of which Is an exact differential;
the exterior differential of the product Is thus zero.
Polncarl's theorem admits a converse, but we do not require It here.
II. DIFFERENTIAL TENSOR FORMS
187. Besides the differential scalar forms with which we have been
concerned until now, there are also the differential teneor forms.
Firstly we restrict our attention to a Euclidean space with respect to a
fixed system cf Cartesian co-ordinates. Let us consider a p- dimensional
domain of integration, to each element of which we assign an Infinitesimal
tensor; each cf the components of this tensor Is assumed to be a differential
form of degree p. The point In question here Is that for a mixed tensor with
two Indices, each of Its components will be a differential form u^. The

Bianchi Identities
205
geometric sun of all Infinitesimal tensors In question will be a tensor of the
sane kind with components f^/·
It Is possible to make an exterior differentiation of a tensor
differential form where the exterior differentiation of ω/ Is <£>^.
188. If Euclidean space Is considered with respect to arbitrary
curvilinear co-ordinates, there exists a natural Cartesian franc attached to each
point In the space. To obtain the absolute exterior differentiation of the
contravarlant vector form ω It Is possible to Introduce a uniform vector
field with components X. and consider the sum Χ-ω1; the exterior dlfferen-
tlal Of this sum, which ie a ecalar quantity, will be
X^ + dXj^ - X^ ((£? + [ω^Λ) .
The absolute exterior differential In question will then be
(6) [tf - dC* + [ω£ Λ . (6)
In the same way, we shall obtain
(7) Duu - <£J£ - [u£ ϊ^] , (7)
and more generally, for a tensor form u^ with two Indices,
(8) Dto/- Α»/ - ω£*ϊ£ +ω*'ω* · (8)
Me add, that as In the case of ordinary tensors, the absolute differential of
a product Is obtained by applying the product rule of differentiation, but here
replacing ordinary differentiation by absolute differentiation.
For example, we have
0[Jkduk] - [Da/fciufc] ,
where ρ Is the degree of the form ϊ^.
189. Let us now turn to a Rlemannlan space. If we taken an arbitrary
domain of Integration In this space, the geometric sum of an Infinity of
Infinitesimal tensors (e.g. vectors) attached to the elements of the domain of
Integration does not have any sense. But If the dona1 η In Its entirety Is
Inflnlteslmally near to a given point A of the Rlemannlan space, we could
replace the line element of the space by a Euclidean line element osculating
at A, The tensor Integral will then have a sense, and Its principal part,
which Is a tensor attached to A, ie independent of the osculating Euclidean
metric ae chosen*

206
Bianahi Identities
For a particular case, let us take a (p+1) dimensional dowaln Mlth a
p-dlmenslonal boundary. The tensor Integral of the eleeent ω? along this
boundary will be equal to the Integral of the eleeent Du/ across the given
domain. Now at A, the coefficients of Οω? only Involve the osculating
Euclidean metric on account of the coefficients r:j,, which are the вате for
the Riemarmian metric
190. Let us take, for example, the vector Integral (dto across a wry
small cycle. Here we have
~~i j. *
ω ■ au
0=4 " Цл"] " 7 (Tth - «мО1****1 " ° ·
Consequently, the geometric sun of the vector в ffi* whioh join a point of the
cycle to a point infinitesinally near is ыго. This result may be related to
those Ideas developed In the preceding chapter. In fact. It proves that If
the cycle Is developed In Euclidean space, the geometric sum of the
corresponding vectors W* Is zero, and consequently the displacement associated
with an Infinitesimal cycle of any form reduces to a rotation.
III. THE BIANCHI IDENTITIES
191. Let us proceed from the relations as stated In Nos. 160 and 163
which give the forms ω/ or Ω., defining the Rlemaimlan curvature:
ω/ ш Λδ( - [Uifcu# (9)
%' " *« + ^tt-S1 * (10)
Let us make an exterior differentiation of both sides of the equations In (9).
On account of these equations themselves, we obtain the new relation
<»/- - [Q^] +[ω*£)^] . (11)
On referring to (8), we see that the relations In (11) state that the absolute
exterior differential of the tensor differential form Ω? is жего\ this we
shall write as
Dn/- 0 . (IT)
As the form ω/ Is of the second degree, Dfi/ Is of the third and the
relations In (IT) may be expressed In terms of the calculus of absolute

Bianahi Identities 207
*
differentiation In the form
Κ/α0|γ+ Κ/βγ|α+ *ΐ γα|0 " ° <*^·«·β·Υ ' 1-2 n)... (12)
The relations In (12), which are just a representation of those In (IT), con-
**
stltute what are known as the Bianahi identities.
The tensor form п.. Is only the form u. written In Its covarlant
form, Its absolute exterior differentiation Is Identically zero. The relation
which gives the Identities
R.. й| + R.. - , + R., .- - 0 (13)
tj οΒ|γ ίο 0γ|α ij γα|β '
could, moreover, be derived directly from (12). The tensor Ω.. or rather the
opposite tensor -п.. represents the Ыvector that defines the rotation asso-
elated with a surface element of the space. We may straight away deduce from
this and from No. 189 the geometric significance of the Blanchl Identities:
If an elementary three-dimensional domain is ooneideredj the biveotore that
represent the rotations associated with the surface elements enclosing this
voliene have a geometric sum equal to яего.
IV. POINCARE'S THEOREM IN RIEMANNIAN SPACE
192. We have seen (No. 186) that the second exterior differential of a
differential form Is Identically zero; this Is Po1ηcare's theorem. The theore*
evidently extends in Euclidean space to any tensor differential form.
Generally speaking, this Is no longer true In Rlemannlan space.
To exemplify matters, let us start with a vector differential form with
components ω1. Its absolute exterior differential (see No. 188) Is
—t i i — k
0ω ■ dm + [ω. ω ]
We again make an absolute exterior differentiation
D2^ ■ d([tf) + [ω£ эЛ ;
the calculation straight away yields
DZ5* ■ [^€ω*] (14)
*
We recall [No. 41, footnote (1)] that a small vertical bar placed before one
or more Indices Is a symbol of absolute or covarlant differentiation, or of
several successive covarlant differentiations.
**
L. Blanchl, Sui simboli a quatro indiai о Bulla curvature di Riemann [Rendw.
Aoad. Linaei (5), t. 11, 1902, pp. 3-7].

208
Bianahi Identities
The Rlemannlan curvature of the space enters here and In
general prevents the second absolute exterior differential of ω1 being zero-
Subsequent absolute differentiation yields
and
D3^ = [0^ D5*]
D4^ - [Ω*ί£Λ
We could have similar expressions by taking any tensor form.
If In particular ω1 ■ аиг, the absolute exterior differential uu1 Is
zero, the second differential must therefore be zero and consequently we must
have, on account of (14),
[<«Λ£]
(15)
This relation In return gives the equations In (12) (No. 163) which Involve
the components RJ^fc. Rfc£Jt» Кш of №е curvature tensor; it oould he regarded
as a representation of these equations.
Another Interesting application of (14) Is obtained by taking ΰΛ as an
ordinary contravarlant vector field Хъ. We shall consider a cycle С
bounding an Infinitesimal area, I.e. all points of which are Infinitesimally close
to a point A. The integral /dx\ taken across the cycle Is equal to the
double Integral //x аг taken across the area. If then this area Is
equivalent to an Infinitesimal Ыvector p1, we have the relation
I™1 - \ RL ρΓβχ
•'с
Following Note II of No. 158, the geometric variation уХъ due to the
rotation associated with the cycle can be found to be
„vi . 1 «*Dt re
VX " " I X Rkrs p
pie mixed components a°. of the Ыvector representing this rotation Is
therefore
This result Is Identical to that which brought about relation (6) of No. 162,
proved for the particular case where the cycle Is an Inflnlteslaal
parallelogram. We oan now eee that thie result holds for every infinitesimal cyoles
whatever ite form (of. No. 190J.

Bianahi Identities
209
Remark. From the relationship giving the covarlant exterior differential of a
tensor form, we can deduce that this differential is zero If the Ыvector form
к ΐ
of the fourth degree [Ω·Ω7] Is zero. This will also be seen to be the case
If the space Is two or three dimensional. It may be easily verified that this
Is still the same, If for a space of any number of dimensions, the curvature
Is constant.
V. CURVATURE VECTORS AND THEIR FIRST REPRESENTATION
193. Let us return to the geometric Interpretation of the Blanch1
Identities. They express (see No. 191) the fact that If we consider a
three-dimensional element of the space, the geometric sum of the Ыvectors representing
the rotations associated with the surface elements of the boundary of the
domain Is zero.
Here we have been concerned with free Ыvectors. Let us see what happens
If we were to consider bound Ыvectors (see No. 19). To each surface element
there would then be an associated tied Ыvector
The geometric sum of all of the bound Ыvectors gives a bound Ыvector and a
free trlvector, the first Is zero on account of the Blanchl Identities; hence
there only remains the free trlvector. Now the Integral
jfi *»<·,&
evidently yields the free trlvector by absolute exterior differentiation
fffl (dbV* + dJilki * du4j)[e.e.ek) .
Vie shall make a convention of saying that the trlvector with components
^k - шЧк\ * idJuH) * [du4J] . (i6)
or rather Its negative, represents the curvature trlvector of the
three-dimensional element In question. The tensor produced by the coefficients has six
Indices:

210 Bianohi Identities
ijk jk ki xq
rv:j -
Kiht
hVn
It? + R'
«S
0
(*rh)
(ΐ*ΛΜ·« all distinct)
In these relations there Is no summation with respect to Indices twice repeated
which have fixed values.
194. Let us now consider (for η ζ 4) an elementary four-dimensional domain
In the space and the free curvature tr1 vector of the three-dimensional elements
on Its boundary. Their geometric sum coats about by the absolute exterior dlf-
•LiJe ί
ferentlatlon of the form ω . Now this Is zero, since each of the forms du
and ir has a zero differential. Thus the gecmetrio em of the free aurva-
ture triveotore of the elements of the boundary of an infinitesimal four-dimen-
eional domain is zero.
If the hound curvature tr1 vectors are considered, this Is no longer the
case and a 4-vector Is obtained with components
Q^** - [*V**] - idJuikh) + [**Q* ] - [dA*fc]
- 2 {[d*Wi/*]t[^AV*] + Idukdu4h) (17>
+ [<uAbVfcl + ldJduhd*l + [dukduhaij]} .
This i-vector, or rather half its negative, oouli be regarded as defining the
free curvature 4-veotore of a four-dimensional spatial element.
It is possible to continue these operations and define
at each stage the (free or bound) p-vector curvature of a p-dimensional spatial
element. The following general theorem Is thus obtained.
Theorem. Given an infinitesimal p-dimensional domain in a Riemamian spaoe,
the geometrio sum of the free (p - 1) ourvature veators of the elements of its
boundary is zero. The geometrio вил of the tied ourvature (p - 1) veotore of
the same elemsnte is equal (to within a numerical factor) to the free
curvature p-veotor of the domain.
195. Let us see In particular what happens In the case of an Infinitesimal
(n - D-dlmenslonal domain. The (n-1) curvature vector has for Us components
^-^.UV..^»-^]*

Bianohi Identities
211
Here we have
tlt«. ..t - , ΐΐ.
Γ2"·*η-1 . 1 RVB
lV'Vi * Vb
pV2"'V2Vl я pWn-l
гЛ9..Л 9i , ki
1 Ζ n-Z n-1 η
On the left hand sides of these relations, It Is unnecessary to sun with
respect to Indices twice repeated; on the right hand side of the first
relation, the summation Indices i ia take values ΐ,,ΐ4,...,ΐ ,; on the right
α ρ \ с η-1
hand side of the second relation, the sunratlon Index к takes the values
1,2 n.
Let us orient the space and denote by l-do the covarlant components of
the supplementary vector of the (n- l)-d1mens1onal element In question.
Similarly, we denote by q.da the supplementary vector of the (n-1) curvature
vector of the given element.
By putting R ■ r££ (see No. 180), we get
It thus Introduces the Rlemannlan curvature scalar R and the contracted
tensor R.· (see No. 180). These relations could be Interpreted In the
following manner.
In the Euclidean tangent space at a point, let us consider the quadrlc
having this point for Its centre and for Its equation, we have
Vv ■ \R °i/*j - Vv ■] ■
We shall refer to It as the Einstein quadrio. The curvature of the (n - 1)-
dlmer
with
dimensional element of magnitude da could be represented by a vector q.da,
the ft denoting the contravarlant components of the unit vector noma! to
the element. It can he eeen that the vector is normal to the diametral hyper-
plane conjugate to the direction I with reepect to the Einstein quadrio.
The general theorem In 194 then tells us that the geometric sun of the
vectors which represent the curvatures of the elements of the boundary of an
infinitesimal η-dimensional domain is zero. Analytically, this theorem could
be stated by writing the divergence of the tensor S.. as zero, or

212
Bianchi Identities
the ев are, for η = 4, the equations which, in Einstein1 г theory, espress the
theorems of conservation of momentum ("quantity of movement") and energy. The
vector representing the curvature of a space-like, three-dimensional element
(space-time) uniquely represents the momentum and energy of that element.
We note that (19) gives as a particular case those relations In (16)
(see No. 169) derived In representing the three dimensional spatial curvature.
196. The Rlccl principal directions (see No. 180) are simultaneously the
principal directions of the Rlccl cone and the Einstein quadrlc. It Is now a
straightforward matter to prove a general theorem of Rlccl which we have
already considered for the case n-3 (see No. 171).
Let us consider a totally geodesic manifold V ... The normal to this
manifold stays normal when It Is displaced by parallelism along an arbitrary
path across the manifold. Consequently, the rotation associated with an
arbitrary cycle on the manifold leaves this normal fixed. The bivector associated
with such a oycle is thus entirely tangent to the manifold. The Immediate
result Is that the tr1vector associated with a three dimensional element of
the manifold Is Itself also entirely tangent to V _,, since It Is a sum of
simple tangent trlvectors. This argument extends to an element of V , of
any number of dimensions. In particular, the (n - 1)-vector which represents
the curvature of an (n - 1)-d1mens1ona1 element of V . Is tangent to V ,
π—Ι П~ I
and the supplementary vector qda Is normal to V ., I.e. normal to the
element. The normal to V _, Is thus a principal direction of Einstein's quadrlc,
that Is to say of the space.
VI. VECTOR CURVATURES ANO THEIR SECΟΝΟ REPRESENTATION
197. We have defined, In the preceding section, the Rlemannlan curvature
of a p-dlmenslonal spatial element and represented this curvature by a p-vector.
There Is a second kind of representation In terms of the (n-p)-supplementary
vector that can, moreover, be taken as free or hound. This second
representation assumes a previous orientation of the space. We have already utilised
this fact for p-3 (see No. 169) and p « (n - 1) (see No. 195).
If the free (n-p) -vectors are taken, the theorem In No. 194 states
that the swn of the free {n-'p)-veotors representing the curvature of the
elements of the boundary of an infinitesimal (p + 1)'dimensional domain is zero.
It Is worth noting that the geometric sun of the same (n-p) bound vectors is
again zero, this being contrary to that realised In the preceding section.
It will suffice us to prove this for p"4 and η-7. Let
0l" -ia™

Bianohi Identities
213
be one of the components of the tr1vector attached to a four-dimensional spatial
element. The free four-vector representing the geometric sum of the fixed trl-
vectors on the boundary of a small five-dimensional domain, has for Its com-
1234
ponent 0 , the expression
Θ1"4 - [Λ1 Θ234] - Ι*? Θ135] + W«3 0124] - [Λ4 Θ123]
■-7[As67J ·
Now each of the terms constituting ^557 has a factor of either the form
Uj. - g-^ydu or one of the forms IL. (t - 5,6,7). Now the sum
Is zero, as well as the sum
which Is zero on account of (15). This proves that the different components
еда are all zero. Q.E.O.
198. In the particular case, p-n- 1, the preceding theorem states that
the vectors which represent the curvatures of the elements of the boundaries
of an infinitesimal η-dimensional domain ooutd be regarded as a eye tern of foroee
in equilibriim. For η-4, this theorem completes the physical Interpretation
of Einstein's gravitational equations. The vectors which represent "momentum/
energy" In mechanics are, In fact, hound vectors and not free vectors. For
n-3, the theorem can assume the role of a noteworthy mechanical form.
Let A be a point In three-dimensional Rlemannlan space. We attach to
this point a rectangular frame and consider a small domain enclosing the point.
The components pda, qdat rda of the vector attached to a surface element of
the boundary of the domain are of the form (see No. 169)
ρ - Kjja + K12B + Κ13γ
q - K^a + K^B + K23Y
г с Κ31α + K32B + Κ33γ
where α, β, γ denote the direction cosines of the normal to the element.
These relations are Identical to those representing the forces of elasticity
In a continuous medium. We thus have the following theorem:

214
Bianahi Identities
Theorem. If a' 3-dimeneional Riemannian space is imagined to be a continuous
medium euah that the elastic pressure which oats on eaoh element of the surface
ie equal to the vector representing the Riemannian curvature of thie element,
then thie medium ie in equilibrium under the action of ite forces of elasticity,
VII. SCHUR'S THEOREM
199. The above considerations naturally lead us to consider what happens
to the relevant theorems of curvature vectors for a space that Is isotropic at
each of Its points.
In this case, the rotation associated with a surface element reduces to a
Ыvector tangent to this element and equal to the product of this element and
a scalar K. The contravarlant components of this bivector are therefore
-Uij « UdJdJ) .
By stating the fact that the absolute exterior differential Is zero and noting
that of the tensor [duO\P\ Is also zero (see No. 190), we obtain
[dKrfuW] « 0
If ηϊ3, all the derivatives -^-r are zero, consequently К Is constant.
Ъи
We then have the following theorem of F. Schur :
Theorem. If an n-dimensional Riemannian epaoe ie isotropic at eaoh of its
points, then it has constant curvature (n£3).
200. There Is a more general theorem for n-4, attributed to
G. Herglotz relating to spaces whose principal directions are completely
Indeterminate, I.e. for which Einstein's quadrlc Is a hypersphere everywhere.
These spaces are again characterised by the curvature at eaoh point being
constant in the different directions in (n- 1) dimensions. For a space having
this property, the (n-1) curvature vector of an (n-l)-d1mens1onal element Is
represented by an (n-1) vector situated In the sane (n-1) plane, as the
element and proportional to It; Its contravarlant components are therefore of the
form
^-^-hUV2...^] .
The absolute exterior differential of the tensor form so obtained Is zero, and
Math. Ann,, t. 27, 1886, p. 563.
^Leips. Ber.t t. 68, 1916, pp. 199-203.

Bianahi Identities
215
for a tensor H, we have
[iHdu Лаи z ... du "~]] = 0 ,
whence
Jf-0 .
The curvature Η 1s thus the same everywhere.
Furthermore, we have
s«-Чту"»«-*« ·
whence
a/-o {itd)
«ί-ξ «£■*«-■■ ·
where the Index t 1n the last relation 1s not a summation Index.
By making subsequent summations with respect to i, we obtain
R - «(^R-H)
«■4r« ·
The Riemartnian aurvature scalar R is thus aonebmt.
The above theorem reduces to that of Schur for n-3. It 1s not unlike
the theorem 1n hydrostatics 1n which a perfect fluid in equilibrium under the
action of Its elastic forces alone has a constant pressure.
He end with an interesting remark, if the curvature of a Rteraanntan space
1s zero in all the directions emanating from a point in p-dimensions, the
Riemann-Christoffel tensor has all Its components zero at this point. An
exception 1s for р-п-l, in which case the hypothesis readily reduces to
"("2+1> relations
R.. ■ 0 ,
which state that the contracted curvature tensor 1s zero. The spaces for which
these relations are true everywhere have zero curvature but only in the direo-
tiona in (n- 1) dimenaiono. This has a meaning in Einstein's theory for empty
space time, where there 1s no momentum and no energy. We refer to Einstein
врасвв as those spaces for which R.. - 0.

Chapter IX
THE METHOD OF THE MOVING FRAME
MANIFOLDS EMBEOOEO IN A RIEHANMIAN SPACE
I. GENERALITIES
201. Up to now we have used, almost exclusively, the natural frames
attached at each point 1n a given system of co-ordinates 1n space. But 1t
might be more convenient to use locally Cartesian frames that are more
appropriate to the nature of the problems outlined, and not necessarily with
respect to the coordinates chosen: Each of these frames is defined by Its
origin и and η linearly Independent basis vectors
β1'β2""' π
The laws of tensor algebra and analysis do not change by the condition that
every tensor attached to Μ ίβ analytically represented by its components
relative to the frame of the origin M.
In particular the contravarlant conponents of the vector Ш' joining
Μ to a point H' 1nf1n1tes1mally near to Η will no longer be the
differentials du1 of the co-ordinates, but will be linearly Independent combinations
that we shall denote by шг. We shall continue to denote by д.. the covarl-
ant components of the fundamental tensor but noting that the fundamental form
will be
ds ■ g.-ja uT (1)
We can also consider the covarlant components ω. - д..ш of the vector ffl'.
i " ^ к
Finally we shall continue to denote by иг. and ω.. ■ ff.-,,ω. the forms
which permit a definition of the absolute differentials of the basis vectors
V
"·< - -ϊ·* · «)
These forms are defined by two types of conditions:
1° Those that arise from the differentiation of the relations
*« ' "id * adi : (3)
217

218 Mewing frame Manifolds
2° Those which express the absolute exterior differential (see No. 188) of
the contravarlant differential tensor form ω , or the covarlant form
ω.,, are zero:
da ■ [ω ω£] (4)
А»£-[ш^] . (5)
It 1s possible, moreover, to go directly from (4) to (5) using the relations
1n (3).
202. The quantities which replace the Chrlstoffel symbols of the first
L.
or second kind are the coefficients γ.,, and γ,, of the forms ω., and
ω., expressed linearly as ω ,ω , ω* :
"if ' учкшк* ωΐ " yik ω* fyj* ' Itflb) (6)
Finally, the fores n? and П.. which define the R1emann1an curvature are
obtained by the same relations that previously occurred 1n Nos. 160 and 163,
I.e.
п.-
/ ■ *■/ - K*^ (7)
«. . ■ tL·.. + [ω? ω.. ]
V V г ok
As for the components of the R1emann-Chr1stofft1 tensor, they are the
coefficients of the forms Ω? or п., expressed as exterior quadratic forms 1n
1 2 п. г ч
ω ,ω , ,ω :
ι к {8)
The relations (4) or (5) and (7) are what we shall call the structural
equations of the space.
203. A particular but simple case 1s that where the frames attached to
different points of the space are all equivalent to each other. Here the
components д.. of the fundamental tensor are constants, and vice-versa. We
then have relations
This 1s also the case, for exanple, when the basis vectors «. are unit and
Ъ

Moving Frame Nanifolds
219
rectangular. This warrants the application to Rieraannlan Geometry of the
method of the moving frame (e.g. the moving trihedral for и ■ 3) as used by
β. Darboux In the theory of surfaces and to a greater extent by G. Rlccl In
studying many of the problems of Rlemannlan geometry.* In this last case the
соvariant components and the contravarlant components of the tensor are the
same and we could write them equally as ω1 or ω. , etc.
204. To determine the components γ. .. (Ricci's coefficients of
rotations) 1n the case of rectangular frames, we shall put
We shall then have, on account of (3) and (5),
yw + yjik' ° · Yifej · 4j* ' qm · <10>
from which we can easily deduce
Ue shall apply this last method to the study of problems 1s Rlemannlan geometry.
II. EXTENSIONS TO THE THEORY OF SURFACES ЕПВЕ00Е0 IN A 3-OIMEHSIONAL
RIEMANNIAN SPACE
205. Let us consider a surface S embedded 1n a 3-d1mens1onal Rlemannlan
space and attach to each point Μ of S a rectilinear trihedral whose unit
vector ω, 1s normal to the surface. Of the six forms ω, ,ω-,ω,,α^ ■
-ω-ρ,ω,, ■ -ω,-,ω,ρ ■ -^ι. the third ω, 1s Identically zero, since every
elementary displacement from the point Μ of the surface 1s normal to ·,.
The equations 1n (5) and (7) are here written as
dm, * [uu ^oi J
doig ■ [(ίή ωΊ2] (12)
^2 " "[ω13 ^З3 " *0 [ω1 ω2] :
* th
Rend. Aacad. Linoei, 5 series, t. 2, 1895, p. 276-332; Rend. Aoaad. Ыпаег,
5th series, t. 19(1), 1910, p. 181-187; t. 19 (II), 1910, p. 85-90. etc.

220
Moving Frame Hanifolds
*ω133[ω12 ^З3 " ΚΊ3[ω1 ω2] (Ί3)
^23* [ω21 ωΊ3] " ^h ω2]
We denote by 1С = -β-,ρι? the Mewnnlan curvature of the ambient space
1n the direction of the tangent planar element to the surface at Μ; K., -
~R1312 d€notes tne m1xed curvature (see No. 174) at Η of the oriented planar
elements contlnlng the vector «,. one being a tangent, the other being a
normal to S. The orientation 1s defined 1n the sense of rotating «1 through
j 1n order to coincide with β. and «. respectively, and finally, IL3 ■
~R2312 denotes a similar nixed curvature.
The equations 1n (12) define the structure of the surface S 1n question
as a 2-d1mens1onal R1emann1an space having as Its fundamental form
ds2 » (Ш])2 + (ω2)2 .
The second fundamental form Φ of the surface 1s given by the relation
Φ ■ -De, Ж = ωιωΐ3 + ^^з
" ΎΊ3Ί(ωΊ)2 + {γΊ32 + W^ + γ232(ω2)2 .
but the first relation (13) shows that γ,,2 * Υ?3Γ ^ sha11 now put
ΎΊ3Ί * αΊΓ ΎΊ32 Β Ύ23Ί " αΊ2 " °2V Ύ22 " α22
As ω._ and ω?, are differential vector forms representing the vector -
De, for the 2-d1mens1onal R1emann1an space constituted by the surface S,
(and similarly for ω. and ω. which represent the vector Ш), the
coefficients α,,, α,ρ, a22 of the form Φ, Independent of the choice of freme
He.i?2, are the components of a symmetric tensor with two Indices. The
numerical value of each of these components 1s determined without ambiguity as soon
as the point Η and the vectors *i.e2 are given.
We shall consider 1n fact an arbitrary curve (C) across the surface
through Μ and tangent to the vector β, at this point (m.de, ω» - 0); the
equality
De] ω^
De1 - ω12β2 + ω13β3 or ж - ж β] + αη·3
shows that α,, 1s the normal curvature of the curve
1 cos V /ΛΑ\

Moving Fr&ne Manifolds
221
V denotlnq the angle of the prlncioal normal with the normal to the surface.
It 1s Heusnier's theorem that asserts the Constance of the normal curvature
for all curves tangent to each other. The geodesic curvature 1s given by
ωΊ2
ϋϋ > J. - sluJt (Ί5)
de pn p "
this varies with the curve, since ω,- 1s not a tensor form. The component
a-2 - Op. 1s the geodesic torsion (see No. 89) of the curve C, as the
relation shows 1t to be:
De- ω?1 De-
*
(Ί6)
Ж ~ Ж βΊ + α21β3 from »"сЬ α12 ' β3 Ж
1 dV A 1
/ι χ — ю =— + —
α12 τ de τ *
9
We know that the principal curvatures J- , «- are given by the
relations Ί 2
αΊΊ"Ί +α12ω2 _α2Ί"Ί*α22^ _ 1
from which
(αη -^)(α22 - jf) -β?2· О
and 1t can be deduced that
all+a22"R7+iq"L
2 1
α11 α22 " a12 *jq
(17)
We thus have an Interpretation of the component a22 as a function of the
elements of the curve (C) and elements of the surface.
With the notations of No. 89, the vector e~ 1s written as -s1n6tl +
cos6t9,e as v. We can Immediately verify with the aid of (16) of this sec-
ά J Da
tlon that the scalar product -·2 -g^ ■ (slnet-j - cos6t2) ^ 1s equal to the
geodesic torsion.

222
Moving Frane Manifolds
η ш 1 + 1 1
*22 ^ 4'~n *
(Ί8)
We note that the third relation 1n (12) gives us a theorem already obtained
(see Nos. 164, 173), the value of du12 being -^.[ω^] where K£ denotes
the R1emann1an curvature of the surface S here regarded as a 2-d1nens1ona1
R1enann1an space; we have then
Ki ' Щ + "o
206. The classic theorems relating to the normal curvature and the
geodesic torsion of the two curves tangent to each other and having as their
basis the tensor character of the coefficients a., of the second fundamental
hi
form, admit generalisations by considering the derived tensors of the tensor
The first derived tensor a... 1s defined by the relations
CW
11
ώΊΊ " ^ΛΖ^ΛΖ "α11 ωΊ +αΊΊ2 ω2
0α]2 - 0α2Ί -ώ12 + Ц1-а22)о>12
" αΊ2Ί ωΊ + αΊ22 ^ " α2ΊΊ ωΊ + *212 ^
^22 " ώ22 + 2αΊ2 ωΊ2 " α22Ί ωΊ + α222 "2 *
If 1t were to be displaced along a curve (C) tangent at Η to the vector
«, (ω, »det ω« = 0), the first relation 1n (Ί9) gives
(Ί9)
Mil
ώΊΊ
ds
d±-
7n Ш12 . P«
'12 ds ds
_ _2_ .
τ
9
. J_
Ρ
9
(20)
the last term of this relation has the same value at Η for all the curves
**
(C) 1n question. This 1s, 1n fact, a theorem of E. Laguerre concerning
the Euclidean case.
Here we dispense with the vertical bar 1n front of the Index of
differentiation, k.
Oeuvres, II, p. 129-130. Cf. E. Goursat, Coure d'Analyse I, 3rd ed.,
Paris 1917, p. 641-642.

Mooing Frame Manifolds
The second relation 1n (19) gives, under the same conditions,
α121 "α211 Β"ώ"+ {αΊΊ " α22> ΊΓ
--k+(i_- L)J_
-ί Ρ ' Ρ
223
(21)
the last term again has the same value at Η for all the curves (C) 1n
question.
We now look at the second derived tensor a,.^. We have
Da
111 " ώ111 ■ {a211 + a121 + α112)ω12 " allll ω1 + α1112 "2
(22)
^121 ° ώ121 " {a122 + a221 " α111)ω12 " α1211 ω1 + a1212 "2 "
In order to derive these equations 1t must be possible to express αΊι2· a,--
and a?21 1n terms of the components a,,, and α·\ο·\- ^ then obtain after
exterior differentiation, the relations
ω13 "α11 ω1 +α12 "2
шзз - α2] ω, + αη ^
We may regard ω... and uu, as ^e tensor components of the vector -De.
and carry out the absolute exterior differentiations which 1n turn give
(see No. 188):
^3 " ^1ω113^ " ^21ω1^ + ^22**^^
I.e. on account of (13),
a112 " a121 B K13
α212 Β α22ΐ Β *23
(23)
By displacement along the line (C), we can derive the following (22),
I.e.:

224
Moving Frame Manifolds
da
αΊΊΊΊ ""3·"" (3αΊ2Ί + K13} ρ" ·
da
Ί2Ί
Ί211
ds
- (2α.
221 " αΊΊΊ + K23* ρ
αΧ
Finally, by replacing α23Ί by ^ - a^, and α]ΊΊ and α12Ί by their
values from (20) and (21) respectively, we obtain
Ί111
л2 \ ώ pff
.-3_(J-.L)
ρ<? "
(24)
d*J-
dJ-
da *g *g
We thus arrive at the following theorem:
ς pn 61 ,Λ ,
5 ~3Γ~ "τ Ρ " 3ί " *23
- Κ„
(25)
Theorem. If α point Μ ο/ α surface that is embedded in a Ъ-dimeneional
space is considered, then the different ourvee traced on the eurfaae achitting
a given tangent at this point have at thCe point the following in ооттоы
the normal curvature, geodesic torsion and the four quantities
da
_L_L
τ ρ
9 Я
<f 2 ,
da η g
1
d2^
d-L
d±-
ds g g ^ da ' ρ ч п '
i
>+(i_.L)_^ + j_
da1" wn da yg [ ds
1
ч
6 1
Λ
"τ ρ " 3 ds ' *23
ьАяге — . — 3 — denote the normal curvature, the geodesia ourvature and
ρ ' ρ τ "
" ff ff
geodesic torsion respectively. The mean curvature of the surface is denoted
by L (the sum of theprinoipal curvatures), K,, denotes the mixed

Moving Frame Manifolds
225
Riemannian curvature in the ambient space of the two planar elements tangent
to the lines C, one a tangent, the other α normal to the surface; finally
by Kp^ ДО denote the mixed Riemannian curvature at Μ of the planar element
tangential to the surface and the planar element normal to the lines C.
The quantitites K,- and K.. are zero if the Riemannian space has
constant curvature or if the direation normal to the surface is a principal
direction in the ambiant space.
III. LINES OF CURVATURE AND ASYMPTOTIC LINES OF A MANIFOLD EMBEDDED IN A
RIEMANNIAN SPACE
207. Let V be a p-d1mens1onal manifold embedded 1n an n-d1mens1onal
Riemannian space. We attach to each point Μ of the manifold a rectangular
frame (R) defined by ρ unit rectangular vectors β,,β-····»β tangential
to the manifold and (n-p) rectangular vectors βΒ+ι·β0+2'"*·βη nonnal t0
the manifold. Ue shall denote by the Latin letters i,j,k,... the Indices
1.2...alp and by the Greek letters α,β,γ,... the Indices ρ+Ι,ρ+2....,π.
By displacement on the manifold, the components ω of the elementary
displacement of the point
here become:
Μ are zero. The structural relations of the space
аы. - [UfcUfci]
о - [Vkai
^αβ " CuWbe] + [ωαΑωλβ] + 1 *αβ*»[ω*ωΛ]
(26)
Let us commence with the case of a hypersurface (р*ч- 1). We have a
generalisation of the second fundamental form of a surface 1n a 3-d1mens1onal
space by considering the scalar product
■4. Λ - ωΐωίη я vufpj
the coefficients
γ. . are symmetric with respect to the extreme Indices by
virtue of the relation stated on the second line of the equations 1n (26),
where α ■ п. The quotient of this form and ds gives the normal curvature
of an arbitrary curve tangent to the direction (ω, ,ω~.... ,ω Ί).
I С π— I

226
Moving Frame Manifolds
The principle tangente correspond to the stationary values of the normal
curvature at a point; the second fundamental form 1s written with respect to
the frame formed by the unit vectors taken on the principal tangents as
Ί 2 Ί 2 Ί 2
Epl +^ω2 + "' + 1Γωπ-1 ·
on making evident the principal curvaturee p— .
The linee of curvature are the lines tangent at each of their points to
a principal tangent. When 1t 1s displaced along a line of curvature, the unit
vector normal to the hypersurface undergoes an absolute parallel displacement
to the tangent to the line and this property characterises the lines of
curvature.
The аоупфЬоНо linee are those which send the second fundamental fom to
zero.
208. If the manifold V 1s p<(n-1) dimensional, matters are more
complicated. There exists (n-p) quadratic differential forms generalising
the second fundamental form of в surface; these are the forms
*x ш " C0*^ ш СыЛа* ш yia3^biibij^ *а ш p+1 "^ '
the coefficients γ. . being symmetric with respect to their extreme Indices.
The asymptotic linee are those which send all the forms to zero; 1t may
happen that these do not exist at all.
The prinaipal tangente wll] be defined via * generalisation of one of the
characteristic properties of the principal tangents of a surface, e.g., the
property by which the absolute differential of the unit vector normal to the
surface, when displaced 1n the direction of a tangent to this surface, 1s
parallel to this tangent. In the general case the tangent will be said to be
principal 1f the absolute differential of an arbitrary unit normal vector to
V has Its tangential component parallel to the tangent 1n question. As the
1-th tangential component of a vector whose first ρ components X. are
zero, 1s the sum Χω. then 1t 1s necessary and sufficient for a tangent to
be principal, that for some α > ρ, the forms ω ι»шд2»····%, are Ρ1"0001"'
tlonal to the forms ω,,ω».....υ when displaced 1n the direction of this
ι ι p
tangent. Putting 1t another way, It Is necessary and sufficient that each of
the quadrio indicatrioee * ■ 1 situated 1n the planar element tangent to
the manifold hes the tangent 1n question as one of Its axes.
Theorem. A tangent ie principal bihen it is an axis oarmon to the (η·ρ)
indicatrioee of the manifold.

Moving Frame Manifolds
227
209. A line of curvature will, by definition, be a line whose tangent
at each of Its points 1s a principal tangent.
An Important class of p-d1nens1onal manifolds 1s characterised by the
property that the (n-p) quadrlc 1nd1catr1ces have p common rectangular
axes at each point of the manifold. For the manifold V belonging to this
class, 1t 1s necessary and sufficient to be able to take ρ unit rectangular
vectors в-,β.,...,β 1n such a way that the form Φ only contains squared
\άρ α
terns; thus the coefficients of rotation γ. . will be zero for ifj and
the form ω. will be a multiple of ω.. The result 1s that each of the
exterior quadratic forms [%ω,_] 1s zero, I.e. those which give the relations
ΎλαΛβΐ " ykajykM * ° (W-1.2.....P. α,β-ρ+1 η) .
Conversely, these ^^ λ) 1ПшР)^шРш Ί) relations (assuned verified) Imply
that the manifold belongs to the class 1n question. In effect, let us choose
new basis vectors e,,e?,...,e 1n such a way that the form Φ +, only
contains squared terns; we thus have γ. ., . ■ 0 for ifj. By now putting
α-ρ + 1, β-ρ + 2 1η (27), we shall have
Yi,P+2.j4p+bj-Yt',P+l,J] " ° UtJ ' 1·ϊ-·*> !
7 7
1f then the coefficients of (ω.) and (ω.) in φ _, are distinct, we
г j' p+1
shall have γ, „.- . - 0. Let us assume for example that the coefficients of
2 2 7
(«ι) (ω-) ...CO in Φ +, are equal to each other but are distinct froe
the coefficients following, then It Is possible to take In the h-ρlane defined
by e, ,e2,...,e. another system of unit rectangular vectors 1n a way thet does
not alter the form Φ _,,, but sends the coefficients v. .« . to zero for
p+l t,p+Z,j
itd (i.J - Ί.2 h).
We shall proceed 1n the sane way for the remaining coefficients. We shall
thus be able to reduce the two forms Φ , and Φ +? slmultaneoulsy to
squared terns only. The same applies to the fom Φ,, etc. fhe equations in
(27) thus give the necessary sufficient conditions for the manifold to admit
*
ρ families of lines of curvature that intersect orthogonally.
210. A particularly Interesting case 1s where all lines of the manifold
are lines of curvature. In this case, 1t 1s necessary and sufficient that all
the forms Φ should be proportional to the as of the manifold. It 1s then
possible to choose normal vectors e +, ,e +2,...,e 1n such a way that the
These manifolds are those with zero Gaussian torsion. For topics deallnq
with the Gaussian curvature of a manifold, and more generally, on the properties
of manifolds embedded In a Rlenannlan space, see E. Cartan. La Gaanatri» dee
espooes de Riemann (Mem, So. math, IX, Chap, VI, pp. 43-511

228
Moving Fran» Hani folds
forms *p+i»*p+2···'·*η-ΐ are identically zero with * being equal to
Ms ; puttlnq it another way, we have
ω. - 0 (α =- ρ + 1,... ,π - 1), ω. * Αω.
ια ъп ъ
Let us determine all those manifolds embedded in Euclidean space that have this
property. He now denote by α,β,... the Indices p + l,p + 2 n- 1. The
structural equations (26) whose left hand side 1s dto. give ΑΓω.ω ] ■ 0;
those whose left hand side 1s da. qlve [4Αω.] ■ 0. The result of this last
statement Is that A is a constant. There are two cases to be distinguished:
1 If A ■ 0, the manifold ie a plane manifold since the absolute
differentiation of the entire normal vector 1s normal to the manifold.
2° If A f 0, the forms [ω.ω ] are all zero: 1f then ρ a 2, the forms
ь not
ω are zero and the equations 1n (26) whose left hand sides are άω
πα ι an
are Identically verified. The point 0 ■ Μ + }e 1s then fixed. Its differ-
A n
entlal being zero on account of the relations ш ■ 0, ω . B -Αω.. The man1-
na пг г
fold 1s a locus of points equidistant from a fixed point 0. On the other hand
the (p + 1) plane containing the manifold V and the point 0 1s fixed since
the differentials at the point Η of the vectors e. and of the vector e
ι η
are 1n this (p + 1) plane.
Theorem. The submanifolde V of Euclidean space, all lines of which are lines of
curvature, are the hype rp lane в and the ρ-dimeneional hyperapkeres.
IV. RIEHANNIAN SPACES SATISFYING THE AXIOM OF THE PLANE
211. Let us mention another application of the method of the moving
frame by proving analytically the theorem that was previously proved by
geometric means (no. 179). We assume that an n-d1mens1onal R1emann1an space has
the property that every p-d1mens1onal goedeslc manifold (2έρ£η-1) at a
point is totally geodesic.
Let us attach to each point of such a manifold V a rectangular frame
as we did 1n the previous section. We then have, on displacement across this
manifold, ω -0. The manifold 1s totally geodesic 1f the absolute
differential of each of the vectors e, 1s tangential to the manifold. Putting it
another way, 1f the forms ω. are all zero (similarly for the forms Φ ).
ta a
The equations (26) show that all the components R·^ must then be zero
(t,M ■ 1,2 ρ, α e p + 1 n). In particular with it j, к now
denoting three distinct Indices taken from the set 1,2 n, all of the
components of R.... are zero; effectively, 1t 1s always possible to envisage
a geodesic manifold V at a given point such that the two vectors е., е,
are tangential and the vector e, 1s normal to it at that point. This shows

Moving Frame Hani folds
229
that the mixed Riemcmnian curvature of two planar elements having a ootmon
line element and perpendicular to eaoh other is aero. We are going to prove
that this property involves the 1 sot ropy of the space at each of Hs points.
212. By an 1nf1n1tes1na1 rotation of the frame attached to an arbitrary
point 1n space, the components R, ... undergo an 1nf1n1tes1na1 linear
transformation. Now, by a simple 1nf1n1tes1na1 rotation through an angle a,
parallel to the biplane determined by the basis vectors «., β , the compo-
v η
nents Χ., Χ of a vector undergo the elementary variation aX and -aX.
к т я J.
respectively, the other components do not alter. Through such a rotation, that
we shall denote by the symbol [Vn)t the quantity R..j., which transforms
as the product X. Y. Z. T. of four vectors, will undergo an equivalent ele-
*- J К fl
mentary variation up to a factor α as the sum of the components obtained
by replacing (wherever 1t occurs) the Index t by m, minus the sane sum
but here replacing m by l.
Ue then apply the rotation {j k) to the component R---fc which must
be zero by hypothesis, regardless of the choice of frames; we shall then have
RtJv ш Rikik *w1th su""""10")· (28)
Let us now apply to the same component (R....) the rotation {it}
(t t i,J,k) and the result 1s
Rfjik + RiJVc'° i
by a cyclic permutation on the Indices iM J, k we obtain
D ж Q ж Q
КУИ ikij "tijk *
but as the sun of the three terms of this double equality 1s zero, each one
1s then zero.
The only components of the Rlemann-ChHstoffel tensor that are not zero
are therefore out of necessity of the form R..... But owlnn to (28) these
components preserve their numerical value by the change of any one of two
Indices because they all have the same value - K. The space 1s therefore
Isotropic at each of Its points.
The proof shows that 1f all p-d1mens1onal geodesic manifolds at a point
are totally geodesic for a particular value of p<2, then 1t 1s the same
for all the other values of рг2 (cf. no. 115).

Chapter X
NORMAL RIEMANNIAN COORDINATES
I. NORMAL COORDINATES
213. We shall consider a point Ο 1η R1emann1an space and, as 1t 1s
always possible, a rectangular frame R 1s Imagined to have this point as
Its origin. Every point N 1n the space, sufficiently near to D« 1s on a
geodesic stemrlng from 0. Let a. be the direction cosines of Its tangent
at the origin and s the arc OH measured on this geodesic. The normal
coordinates of Μ are the η quantities хг defined by the relations
χ ■ α β (1)
It 1s nearly always possible when transfer1np from any one coordinate system
to another to have at 0 all coordinates zero and the coefficients д., of
the fundamental form equal to 1 for i-j and 0 for ifj. It suffices
to carry out a suitable linear transformation with constant coefficients on
the u.. Let us assume that this has been done.
We shall obtain the normal coordinates by Integrating the geodesic
differential equations, viz.
w2 <
, i du du я n
л« к h ~αΤ ~αΤ υ
with the Initial conditions that и1 - О for e-0. Ue shall then have
хг
214. The relations 1n (1) define, as we might say, a representation of
R1emann1an space on a Euclidean space 1n which the χ 's play the part of
classical rectangular coordinates. It may be seen Immediately that a geodesic
steering from 0 1s represented by a straight line and a geodesic manifold
at 0 by a hyperplane. The Euclidean space on which this representation 1s
made will be said to be the normal Euclidean space attached to 0.
It 1s easy to see that the normal Euclidean space osculates the Rleman-
nlan space at 0. We have then,1 η normal coordinates, the qoedeslc
differential equations.
2 i . , к , .
a x _^ т. ъ ax ax
*h\w w'° * <z)
231

232 Normal Riemannian Coordinates
They are true if χ by α β where the аг are arbitrary constants. He
thus have at every point of the geodesic 1n question,
ГЛ-*■*-» ·
In particular, at the point 0, the relation holds whatever the constants a\
We thus have
ш. ■
= 0
о
which states precisely that the normal Euclidean space 1s osculating the
Riemannian space at 0.
2
215. He now endeavor to calculate the da of the Riemannian space 1n
the neighbourhood of the origin 0. By putting
ч2 / o\2 / _v2
ώ2.(ώΊ)% (ώ2)....(ώ") .
2 2
the difference da - da must have all Its second order coefficients at least
i °
with respect to χ . Let Ф(х,ая) be the set of second order terns 1n the
2 2
development (assumed possible) of the form da - da . On a geodesic steaming
2 2 ° i.
from 0, we have da ■ da ; moreover the dx1 are proportional to the χ .
We have therefore
Φ(χ,χ) = 0 .
In consequence we presume that the form Φ(χ,αχ) 1s homogeneous and of the
second order with respect to the quantities хгсЬ? - x^dx1. This 1s what the
later calculations permit us to show.
II. THE FUNDAMENTAL DIFFERENTIAL EQUATIONS
216. The frame (R ) attached to the point 0 1s rectangular and we
attach to every point Μ sufficiently near to 0 the rectangular frame (R)
which 1s determined by the original frame (R ).
This being the case, we are going to calculate the forms шг- ω., ω..«-ω:,
which define the Infinitesimal translation and the Infinitesimal rotation which
carries the frame attached to Μ to that attached to a point И
infinitesimal 1 у near. But rather than express these forms in terms of normal coordinates
x\ we shall firstly express them 1n terms of (n + 1) super abundant
coordinates a\cι,...,α", t such that хг - at. The coordinates α stay
constant along a geodesic Issuing from 0 and are the direction parameters of
this qeodesic at 0 with t being zero at 0. It 1s quite evident that the

Normal Riemannian Coordinates
233
forms ω., ω.. only depend on the products at - χ ; 1t 1s possible to
г г3 i i
replace fc everywhere by 1 and a by χ 1η order to express them 1n terms
of the хъ.
If we leave the coordinates a1 fixed and varying t alone, the frame
(R) undergoes a parallel transport and for an equal value of dt, the
vectors Ш stay equivalent to each other and consequently preserve the same
components which are actually ul ■ a%dt\ as for the ω., they are evidently
zero under the same conditions.
He now denote by ш1 and ω., the expressions that take the form ω.
and ω., respectively, when t 1s fixed but the a are variables; they
12 я
are linear 1n da ,da s...tda whose coefficients are functions of t and
The complete forms ω , ω., are given by the forms
ω (t,ajdt,da) ■ a dt + ω (t,a,da)
ω. ,(tta,dttda) ■ ω. .[t,a,da)
(4)
217. We are going to see how the forms ω1, ω1 , regarded ав functions
of fc, are determined. The role of the quantities аг, dab 1s
that of parameters for a system of first order linear differential equations,
Let us start with the structural equations [see nos. 201 and 202 relations
(4) and (7)].
ώα
Г к Ί
*Υί " ^ΐΑί1 + \ RvfcA ^*1
(5)
By repladnq ω1 and ω., by their equalities In (4) and displaying 1n the
resulting equations the terms containing dt, we obtain
[dd-dt] + [dt %-] + atf - [(aKdt + ϊϊ6) ω..]
dt
fct
(6)
[^ -g£4+<££ - сг£а + J r.>Wi [(At + a*) * (At + ε*)] ·,
we denote by <£?', dm.., the exterior differentials of the forms HP", ω.,
where t ie a parameter. On equating terms containing dt on both sides of
(6), we obtain the required differential equations which are fundamental,

234
Normal Riemannian Coordinates
3ω _ , i , k-
9ω.
-¥
Τ · \ hm{a^ -aV£)
RijWi a ω
(7)
We have to Integrate these differential equations. The Initial conditions are
clear, for 1f t ■ 0, the frame (R) 1s, for all values of
the fixed
frame (R ) attached to 0 of the type that whatever the values of a1 and
i
da . we have
ω (0,a;da) = 0 ω., (0,α;<2α) - 0
(8)
For the form ω да obtain the system of differential equations
Э ш _ D к h~i
—£ R^ ααω
(9)
with the initial conditions
~ΐ - η fru _ ,i
ω - 0 , "g^- da
for t ■ 0
(10)
218. The differential equations in (9) give rise to the following
important result:
Theorem: If the fundamental form ia given αβ α function of the normal
coordinates relative to an original frame (R ) with the componante of the Riemann-
Chrietoffel teneor relative to the family of frames adapted to these coordinates
and is expressed in terms of the normal coordinates in question, then it is
uniquely determined.
In fact, as the components Rf.-M are assumed to be continuous functions
of the coordinates t, a ta ,.,.,α*, the equations in (9) admit a uniquely
determined solution if the Initial conditions are taken Into account. On the
other hand the forms ω1 on account of (4), are derived from the forms ω1
on taking t-1, аг'хг. The de of the spece 1s thus determined exactly.
We are able to add an Important corollary:
Corollary: Given two Riemannian spaces Ε and Ε' of equal dimension and
in each of them, two systems of normal coordinates relative to two rectangular
frames (R ) and (R') at the origin, and given the components of the

Normal Riemannian Coordinates
235
Riemann-Christoffel tensor, with respect to the family of frames adapted to
the coordinate system in each of the spaces, are the same functions of normal
coordinates, then the two spaces are isometric to each other*
In fact, the two fundamental forms have for their coefficients, the sane
functions of normal coordinates. The map 1s local, for the theorem only
makes sense in a neighbourhood of the origin such that throuoh every point of
this neighbourhood there passes one and only one geodesic stemming from the
origin and not leaving that neighbourhood.
219. Let us now consider en analytic space. This means that it 1s
possible to choose a coordinate system u1 s.t. the components д.. of the
fundamental tensor are analytic functions of the coordinates. In this case,
the normal coordinates x1 relative to the arbitrary frame Ro at the origin
are analytic functions of the и1 and the components R..fcfc of the Riemannian
curvature with respect to the family of frames adapted to these x1, are
equlvalently analytic functions of the хъ and consequently of the coordinates
t and a1. They are thus completely determined 1f at fO their numerical
values are known as well as those of their successive derivatives with respect
12 и
to t. Now on varying t and fixing α ,α ,,,,,α, with the frame (R)
displaced by parallelism the ordinary differential ^--^ taken with respect
to t 1s Identified with the absolute differential; we then have at every
point
9RtJfc*
9t ij
R,,,.,y* a
W
92R
d Kijkh . D /t £ m
-jf-я 'W'm * a ·
We thus arrive at the following theorem.
Theorem: The de of an analytic Riemannian space expressed in terms of
normal coordinates relative to a point 0, is completely determined if the
numerioal values of the oomponente of the Riemann-Christoffel tensor are
known at 0 as well as all their successive covariant derivatives.
To this theorem, there corresponds in a similar way a sufficient
condition for mapping two analytic Riemannian spaces of equal dimension.

236 normal Hemannian Coordinates
III. THE do2 OF SPACES WITH CONSTANT CURVATURE EXPRESSED IN NORMAL
COORDINATES
220. On account of the theorems in 218, all R1emann1an spaces with given
constant curvature К and of equal dimension have the same fundamental form
with respect to an arbitrary system of normal coordinates. All these spaces
are thus Isometric in an Infinity of ways, for once the rectangular
frame (R ) 1s chosen in one of the spaces, it 1s possible for it to corre-
o
spond to an arbitrary rectangular frame in any one of the other spaces. Me
also recover, by the same line of reasoning, the property of free mobility of
spaces with constant curvature.
2
To effectively calculate the da in normal coordinates, we need to
Integrate the equations in (9). Firstly on account of the first equations
in (7) and the antisymmetry of the form ω.,, we see that the sum α ω,
г г "
admits the sum α da on differentiation with respect to t. We then have
α ω. ■ ta da
By putting
(.-?. иг....
we shall have
α ω; ■ tldt
♦(·")'· >г
(ΊΊ)
Let us now note that the equations in (9) take the form
Э ω1 „ к , k-i i г~к\
—=— * ■ Κα \αω-α-αω) ,
from which we deduce
b2(ai 13? -Jrf) v к i , ft-?* J-к, . „ к J, *-t v-*,
—■ ч «- я - К a a [a ur - <τω ) + Κ a cr [α ω - α ω ;
Η
■ - К I (α α? - ύτω )
As the Initial value of the derivative of aV-Λ1 with respect to ι 1s
a da3 -cPda1, we can straightaway deduce by calculation that

ЯогчпаЪ Rimannian Coordinatee
237
[ or -ίτω
IK
(ΊΖ)
51гй(сЛГГ) ,^-Λ**) (К < 0)
By taking the sum of the squares of the n'n*" ' equations in (12), we obtain
the relation
КГ
2 2 2 2
- t2 [(ω1) +(ω 2) + ... + (и?1) ] - t2 [a]da} + ...+andan] .
(Ί3)
The sunmatlon 1s extended over all pairs of Indices itj ■ 1,2....9η.
221. To obtain the de in terms of normal coordinates, it 1s sufficient
in (13) to replace t by 1 and a1 by x1 and see that it becomes the sum
2 2 2
(ω1) + (ύ?) + ... + (L71) . Ue thus obtain
',1,1, 2.2, a «j «\2
[x ax + χ ax + ... + χ ax )
da'
ΐη2/κ[(χΊ)2+(χ2)2+... + (^)2] ^ . , , -3
κ[(χΊ)
(χ")']
1 2 ? 2 и :
(.') + (xZ) + ... + (xn)
(Ί4)
But it 1s also possible to obtain the form of da in polar coordinates by
12 η
subjecting α ,α ,...,α to the condition £-1; t then becomes the radius
vector 0M that we shall denote by r.
Ue can straightaway deduce from the equations in (11) and (9) that
Ξ* - Si"(r *> da1
r Λ
from which

238
Normal Riematmian Coordinates
and by a straightforward calculation,
7
ώ2 я ^2 + slnW) ^2 (K > 0) (15)
К r*
or
ώ·2 - s1nft2 ^) (к < o) .
К Г
2 1 2 2 2 я 2
on denotino by da the sum {da ) + (da ) + ... + {da ) , the fundamental
form of the hypersphere of radius 1 of (n-1) dimensional Euclidean space.
222. It 1s possible to express the relation in (14) under another guise
that 1s obtained by replacing (xW + x2dx2 + ... xndxn) by
2
r2d8Z - Σ tfdJ-Jdx-)
0 iJ
which gives*
ώ2+ώ2 . К,2 - s1n2(r Л) Σ (х^·-^)2 ,
К г4 (tf)
2
where da denotes the fundamental form of the Euclidean space with respect
to rectannular coordinates χ ,x ,...,x", and r denotes, as in the preced-
1 2 2 2 η 2
1ng number, the square root of (x ) + (x ) + ... + (x ) . Moreover, 1n a
neighbourhood of the origin
Kr2 - s1n2(r Л) я Ί K 2 „2 2 . 27 „3 4 29 „4 8 .
fx зк"?5к r *шк v ~m K r + ■■·
and this relation holds whether К 1s positive or negative.
The relation in (16) confirms, for spaces of constant curvature, the
2 2
suqgestion made in no. 215 for the form of the expression da - da . It
1s this suggestion that we Intend proving for the general case.
IV. PROPERTIES OF THE FUNDAMENTAL FORM IN NORMAL COORDINATES
223. Ue are going to prove the following two theorems.
Theorem I. The form ω - t da is the linear combination of the expres-
siona α da - a da .
*This result 1s that of E. Beltrami, Annali di Hat, 2nd series 1869, p. 243,
relation (20).

Hormal Riemannian Coordinates 239
2 2
Theorem II. The form da - da is α quadratic form with reepeot to the
к h h к °
expressions χ dx - a dx .
Let us start with Theorem I. The fundamental differential equations in
(9) of 2Ί7 give
-Ц. (? - t dJ) - RHhj akah if . (9)
Let us establish a priori
s1- * *l ■ Aifcft afcdaft (*ш ■ -W ■
The equations in (9) are true if the unknown functions A... satisfy the
equations
ЭАг«.
with the Initial condition AiW| ■ 0, -^- - 0. These equations admit a
determinable solution, but it 1s necessary to show that the functions obtained
are antisymmetric with respect to the last two Indices. Now this 1s evident
because the functions A... + A... along with their first derivatives at
гкп thk J
t-0 being zero, satisfy a system of linear homogeneous equations. The
theorem 1s thus proved.
о
224. The da of the space 1s the sun of the squares of the expressions
ω1 when t 1s replaced by 1 and a1 by χ . Now the relation
ω* ■ t del· + AifcA akdah
shows that the proof of Theorem II reverts to proving that the sum A.,, da"
1s a linear combination of the expressions avda8 - a8dav. Ue thus establish
A... da1 - Bb. a?da8 (B,. - - В.. - - В.. )
ikh khre x khrs khar Were'
These relations will be compatible with the equations in (10) if the functions
BWire alon9 w1tn the1r first derivatives at я"0 belnq zero, satisfy the
differential equations
9 BWir« _ . η . i
^r5- " t R .. + α R .. A...
n1·
and it is clear that the solution of these equations complying with the

240
Normal Ri&narmum Coordinates
Initial conditions 1s antisymmetric simultaneously with respect to the
Indices r and в and with respect to the Indices к and ft.
225. We are now going to seek an expansion United by the forms or as
well as the fundamental form. For this we determine the expansions of the
functions A.%. and B,, up to the terms in t assuming of course that
the fundamental tensor admits continuous partial derivatives of a sufficiently
high order. An easy calculation yields on account of (17) and (18),
δ μ Ί +3 г D . Ί .4 г я D
\kh Ϊ * а RHkh + ΤΙ * α α Rr€kh/s ■
BWire = 6 * RreWi + ΤΙ * α RrefcVu
Naturally it must give the components of the R1emann-Chr1stoffe1 tensor and
the numerical values at the origin of their covariant derivatives.
Deduced in the development 1s
1? - t dj * \t3 a" frHkh *\ta5 R^laW ,
from which
»* " ^ + l'" <»*!№ + l·" W.> *kdJi ■ (19)
2
He shall obtain the ds by calculating the sum of the squares of the forms
ω1 and consequently putting t = l, al-xl. Не shall obtain
**βΓβ GW,r* 1s deduCed f™ the e*Press1on Bfchre + Br*kh + AtW. Αΐ*ι by
replacing t by Ί and α by xl. The Ί 1m1 ted development of ^J(hrg 1s
3 (RreWi + Ί χ" RrefcA/u} ·
*
from the limited development is
*2 ·*»1*\ «ν**♦ l·" Wu> ***** · <20)
or alternatively,
*Th1s development, exclusive of the term Inside the brackets viz. j x"Rrefc^/u»
1s that of Rlenann (Gessarrm. Werka, Leipzig, 1876, p. 261). It was proved
for the first time by Dedeklnd, 1n his Notes 1n Gessan. Verka da Нвпагт
(pp. 384-391).

Normal Riemcomian Coordinates 241
*2" d*l+ τ? «„» + \ *u KswJ ^ - *W)
χ {xkdxh - xhdxk) . (20')
226. Deduced from (20) are the limited expansions of the components of
the fundamental tensor with respect to normal coordinate в
1 1 * t β II for г = j
9** ' *u + ¥**ъл + 5 * "««/#>*** «·,- - (21)
4j ij 3* ОДЛ 2 * "ikjh/V~ * wij
0 for t/j
as well as the Chris toff el symbols of the first kind, which by first
approximation, are
riW - ■ 1 '»,■№ ♦».««) ** ■ <22>
The symbols of the second kind Г. .» to the same approximation, have the same
expressions on account of the properties of the g , to this approximation,
being equal to Ί for г ■ j and 0 for г f j.
He are now going to apply these limited expansions in (19) and (20) to
the study of the modifications «rising out of the Rlemonnlan curvature of the
space to lengths measured in the neighbourhood of the point 0 in Rietnannian
space and in the osculating Euclidean space normal to 0.
V. COMPARISON OF DISTANCES IN RIEMANNIAN SPACE AND IN THE NORMAL OSCULATING
EUCLIDEAN SPACE
227. He can make a geometric Interpretation of the relation in (20). In
the normal Euclidean space with rectangular coordinates x1. He shall
consider an Infinitesimal parallelogram OMH'P having 0 as one of Its vertices,
Let x1 be the coordinates of M, ахг those of Ρ and consequently хг + ахг
those of M1. The parallelogram defines a bivector with components
Ρ
>3
:W - JdJ-
The relation in (20') states that da , the square of the distance MM', mea-
sured with the Rietnannian metric, 1s equal to ds> the square of this same
distance measured with the normal Euclidean metric augmented by the quantity
Ti RreWi РГвР,Л' "°* follow1n9 the same definition of the R1emann1an
curvature К of the space in the direction of the planar element of the parallel
ΟΙ 2
gram [see no. 174, relation (19)], this quantity 1s equal to - -= К d$ , on
denotlnn by <iS, the area of the parallelogram. He then have the formula
do2 = daZ - i К dSZ .
о
I

242
Hornal Немтпит Coordinates
Let ft be the distance from 0 to MM'; we have
dS ■ bde
and consequently,
ώ2 - ώ2 (Ί -\ К ft2) ,
о J
where
da - da (1 -^ Κ ft2) . (23)
It can be seen that if the curvature К 1s positive, the representation in the
normal Euclidean space produces the length of traced curves to the neighbourhood
of 0. If К 1s negative, the lengths in contrast, are diminished.
228. We are going to treat the relation in (23) more rigourously and
more precisely.
Let us take through 0 in the R1eaann1an space a geodesic segment OH
of length α and let x1 be the normal coordinates of Its extremity.
Following this we take through Η a geodesic segment ИР of length b and suppose
that the unit tangent vector at Η to this segment proceeds by parallel
transport along ON from the unit vector stanning fran 0 and with components
аъ-, finally let уг be the coordinates of P. We, are going to compare the
length b of this segment HP, such that it 1s measured in the Riemannian
space with the length
bo -/V-j,1) + ... + (*"-*">
as it 1s measured in the normal Euclidean space.
Let us firstly evaluate the components X1 with respect to the natural
frame attached to Η relative to the normal coordlantes of the unit vector
resulting fro» parallel transport along OH from the vector α stemming
from 0. The components ω of the vector α with respect to the
rectangular frame RQ are equal to аг; the components ω of the second vector
stemnlnq from M, with respect to the rectangular frame adapted to the system
of normal coordinates, are also equal to α but on account of (19) they are
equal, up to third order Infinitesimals, to
ω " X + Ζ Rrikh xxX »
*In fact in (19), the dxk could be regarded as components with respect to
the natural frame of a vector whose components, with respect to a rectangular
frame R, are equal to ω1.

/formal Rutmatmian Coordinatea
243
this results in the required relations
1 r
5 Krikh
X£ ■ a* - i IT,», ЛУ . (24)
Let us now proceed to calculate the coordinates уг of the point P. The
geodesic MP 1s determined by Integration of the differential equations
with the Initial conditions уг ■ хг for β - 0. We deduce then that
,*■«.* + *¥* Ί ь2 (γ \ νΜ 1 ьЗ /arfc h\ Yfc JiYr
у ■ χ + ьх - у ь (^^ xx -?ь i-^T"jM x x x ·
neglecting fourth order terms. Now let us replace the X by their values
(24) and note that up to second order Infinitesimals» we have on account of
(22),
<ГЛ>Н B " Ϊ <«№ + W *" ·
and that up to first order Infinitesimals, we have
(тг) ■" ι "W+ W ·
Ча» η
Up to fourth order Infinitesimals, this results 1n
+ 6 b * {\ihr + *Нкг> a a
♦ w *3 <W ♦ W ·*Λι> ■
But because of the antl-symmtry of the components R. .., with respect to the
first two or the last two Indices, the last two sums of the right hand side
are zero. As for the second tern of the right hand side, it could be written
as
1 .2 в ι τ i i rw в ft h 8\
"17 rtaft * " α * ) (a * " α * )
1 ? 2
it 1s equal to + ^b Kft , where К denotes the Riemannian curvature along
the planar element containing the directions OH and MP and ft the distance

244 Normal Rlemannian Coordinateq
fron 0 to HP. We extract the relation
bl ш *>2 <Ί + I ***> (26)
or Identically, (on taking square roots)
ъ - bo{} -1 кй2) ,
in almost the exact notation to the relation in (23). Q.E.D.
VI. THE PARALLELOGRAMOID OF LEVI-CIVITA
229. Lev1-C1v1ta refers to a paralUlogramaid as the figure obtained by
taking a geodesic arc AB and another such arc AA'; fron the latter a
parallel transportation 1s made along the geodesic AB to BB1, then finally
there 1s a geodesic arc joining A' to B\ If α 1s the length of AB, b
the lengths of AA' and BB' and a' the length of A'B', there exists an
Important relation giving the Riemannian curvature К of the space in the
direction of the parallelogramold (assumed Infinitesimal) in terms of the
lengths α and a' and of the area S of the parallelogranold.
a
0
F1g. 31
To obtain this relation, let us relate the figure to the normal Euclidean
space at the centre 0 of AB. We denote by a1 the direction parameters
of the unit tangent vector at AA' at the point A, transported by parallel-
Ism once from A to 0. The relations 1n (25) give the normal coordinates
y1 of B' on replacing the χ by the normal coordinates of B. The normal
coodinates y1 of A' are deduced by changing хг to -x^.
The square of the Euclidean distance a' between the two points A1
and B' qives by a similar calculation to that of no. 228
on neglecting fifth order Infinitesimals. Let us assume that

Normal Riemoomian Coordirtatee 245
x1 - | , x2 = 0 , x3 - Ο ,...,
1 Ζ 3
α - cos φ , α * sin Φ , α = 0 , .... ;
we shall have
α'2 = α2(Ί Λ Kb2 sin2 φ) ,
ο j
and on account of relation (26)
a'2 - a2 (1 -f Kb2 sin2 φ)(1 -\ YbZ)
where h denotes b sin φ. Consequently.
a'2 - a2(l - K*2 sin2 ♦) - a2 - KS2 .
*
He finally arrive at the formula given by Lev1-C1v1ta
-2 . r'2
τ
Κ » α V (27)
VII. GEODESIC TRIANGLES
230. Using normal coordinates will allow us to complete the preceding
theorem (see no. 166) on the sun of the angles of a geodesic triangle. He
shall consider an Infinitesimal geodesic triangle ABC In R1emann1an space
and make a representation of 1t In the normal Euclidean space at С (fig. 32)
By calling a, bt о the R1emann1an lengths of the three sides; a , b , ο
the Euclidean distances of the vertices taken 1n pairs, we have
a * a , b ■ b ,
о о
*T. Levi-С 1 vita. Hozione di ρ arable lismo in una varieta qualunque {Rend. Cira
mat Palermo* t. 42, 1917, p. 291).

246
Normal fHemannian Coordinates
then on account of (26)
h denoting the height from C. Now we have
2 2 2
ο = α + b - Tab cos С
о
from which
ο1 = α2 + b2 - 2ob cos С - \ th2o2 ,
ο2 = α2 + b2 - 2ob cos С - J K&ib sin С ,
on denoting by S the area of the triangle. This relation should be
rewritten as
c2 * a2 + b2 - 2ob cos (C - ψ)
(28)
This leads to the following theorem, which 1s classical 1n the theory of
*
surfaces:
Theorem: If on an Buolidean plans, the reotilinear triangule having the вате
агава оа the geodeaia triangle га oonstruoted, then the angles A . В , С
л О О О
of thie triangle are obtained by subtracting the value 2 KS from the
angles A, B, C, of the geodeaia triangle, where К is the Riemannian
curvature of the epaoe in the direction of the planar element of the triangle.
23Ί. The relations
A-^KS
В - В - ^ KS
о 3
C-^KS
(29)
gives, on addition,
π-A + B + C-KS
from which
*0arboux, Theorie dee Surfaces, t. Ill l_1vre VI. Chap. VI.

Normal RiemamtUm Coordinates
247
к = A * γ c" ff (30)
a relationship already proved 1n no. 106.
232. It 1s possible to derive from the above result another proof of
the theorem of Levi-C1v1ta on the parallelopramold by decomposing 1t Into two
geodesic triangles. It will satisfy us to prove by this means a relation due
to F. Sever1 that 1s similar to that of Lev1-C1v1ta.
We shall consider an Infinitesimal geodesic arc AB - a. Let us elevate
from A a perpendicular geodesic arc AA' of length b and then project
from A' a geodesic A'B' perpendicular to the geodesic stemming from В
which results from parallel transport from the geodesic AA' along AB. We
thus obtain a geodesic quadrilateral 1n which three angles are right angles.
I.e. the angles at А, в and 8'.* Let a1 and b' be the lengths of the
arcs A'B' and BB\
Let us take the diagonal A'B and construct 1n an Euclidean plane the
two rectilinear triangles ABA', В В'А' having the same sides as the two
о о о о о о
corresponding geodesic triangles (fig. 33).
F1g. 33
By denoting the area of the quadrilateral by S. we have
ko " 2 ■ Τ
Bo 2 6
_■>
The evaluation of angle AH' 1s not quite so straightforward because the
I ° ° ° /\ w
sun of the angles ABA' and A'BO' Is not strictly equal to ABB' - « and
the three geodesies BA, BA', ВЭ' are not necessarily tangential at В to
the same planar element. However, we shall prove to the degree of
approximation 1n question, with everything 1n accordance \f this were the case, that
*Th1s 1s Lambert's parallelogran of non-Euclidean geometry.

248
Nomal Riemarmian Coordinate*
'ч я τι _ KS
Ό'ο'ο
A.B.B1 " f " χ
Effectively, we shall consider the normal Euclidean space at B, taking as the
1 2
axis of χ the direction BA and for χ the perpendicular direction BB*.
On account of (25) the normal coordinates of A* are
Ί 3 1 ^ ,2
χ = a - * Κ ώ
2.,1„ 2,
the other coordinates being third order. The relations
Ί
cos ABA*
χ
/(χ1) + (χ2) +
(*') + (xc) + (χ3) + ·"
^ x2
cos B'BA' *
/T1—Г2—Г2
/(x1) + (xZ) + (x3) + ...
yield, on neglecting sixth order terms under the-root·
^ч Ί ^\ 2
cos ABA' * * , cos B'BA - *
/(x1) + (x2)2 /(x1) + (x2)
from which we can deduce that the sum of the angles ABA' and B'BA 1s
equal to ϊ , up to Infinitesimals of the fourth order.
Having attained this point, let us project the contour A A^B^B on
Α Β 1η the diagram. If we observe that the angle between the two directions
00 KS
А В and A'B1 is equal to 4?-, we have
о о о о с
up to fourth order Infinitesimals. The two relations
a - a1 ■ b -к- »
a + ar s Za ,
both exact to fourth and second order Infinitesimals, yield on multiplication

Normal Riamarmian Coordinates 249
η1 ν2 . κς2
which is exact up to fifth order infinitesimals.
We then derive Severi's formula*
2 ,2
К - Q "g
^^
identical to that of Levi-Civita. We obtain, by symmetry
КяЪ'2-Ъ2
к Л
VIII. CIRCLES. SPHERES, HYPERSPHERES
233. He are going to introduce some new relationships which in this case
involve the curvature of the space in a planar direction in any number of
dimensions.
Let us outline first of all in a geodesic surface at 0. a circumference
of centre 0 that is defined by taking a constant {very small) length R on
the different geodesies steaming from 0.
It is always possible to assume that the geodesic surface is defined in
normal coordinates by
*3·*4- ... *" -0
He then have on this surface
da2 - (dr1) + (dx2) - 1 K{x}dx2-x2dxh
where К denotes the curvature of the space at 0 in the direction of the
geodesic surface in question.
The area a of the circle radius R is given by the double integral
Ь 2
Jjf/gdxdx'
Now we have
*F, Severi, Sulla aurvature delte euperfioie e varUta (Rend. Circ Mat. Palerme,
42, 1917, pp, 227-259). For the proof, the author restricts his attention to
2-dimensional spaces.

250
Normal Riememn-ian Coordinates
1 1 '
1 „ 1 Ζ
3 Kcx
Ί „ Ί 2
3 ***
Ч*С)
- Ί
Кг'
where г Is the distance from the point to 0. We thus have, on applying
polar coordinates.
A " //(1 - \ **Z)r *
dQ
πΡ1
^ KirR4
(32)
Let A be the area of an Euclidean circle radius R. We have
о
* - *β (ί
T7kr2>
or
A R
о
? Τ?
(33)
This relation allows us to define the R1emann1an curvature Κ 1η teres of the
area A of the circle of radius R. Thfs area fs smaller than In the
Euclidean plane when the curvature 1s positive, and larger 1f 1t 1s negative.
The length С of the circumference 1s readily obtained fro» the relation
1n (32). In fact, A 1s a function of R whose derivative 1s precisely C.
c-^-Ik.r3- «„П-^ки2) ,
fro* which
% - * . к
о
(34)
234. Let us now take a 3-d1nens1onal geodesic manifold at 0, that
could always be assumed to be defined by the equations
■ χ
On this manifold, we have
2 2 2 2
ώ2-(ώΊ) + (dx2) + (dx3) +^R2323(*2<ir3-*3<ir2) + ,.,]

Normal Riemannian Coordinates
251
The volume V of a sphere centred at 0 and radius R 1s given by the triple
Integral
fffjg dxdxZdxZ .
extended to the Euclidean sphere of centre 0 and the same radius 1n the normal
Euclidean space. Now we have by naming γ., the coefficients of the for»
2 2 v
da - da ■
о
g ■ Ί + ϊ1Ί + Υ22 + Υ33 " Ί + *(*\*2.*3) .
* being a homogeneous polynomial of the second degree.
In the Integral
jjl /g άχλάχΖάχ* - Ve + 1 jjj *(χΊ tx2^)dx^dxW ,
1t will be sufficient to consider the squared terms of *, the remainder
becoming zero Integrals. On the other hand by symmetry, we have
///(x1) cfcWcfc3
= Jff {:3)2<bW<b3 . ;///r2 <bW*c3 .^,!5
We thus no longer have to calculate the sum of the coefficients of the squared
terms 1n ♦, a sum which 1s equal to
3 (R2323 + R1313 + R1212J
Finally, we obtain
V " \ * π (R2323 + R1313 + R1212,,rR'
Now the quantity (^323 + R1313 + R1212^ reDresents isee no- 193) the
curvature К (per unit volume) of the space 1n the direction of the 3-d1men-
si onal planar element 1n question. We thus have
V - Vo(l - £ R2)
where

252
Normal Biemarxnian Coordinates
It can be seen that the change 1n the expression of the volume of the sphere
that occurs 1n passlnq from Euclidean to R1eaann1an space only depends on the
curvature of the space 1n the direction of the 3-d1mens1onal planar element
containing the sphere.
On polng from the volume to the surface we can easily see by
differentiating that
-2-Τ " Ι (36)
s/ 9
235. The above relations are generalised without any difficulty to obtain
the volume and the surface of a hypersphere radius R traced across a geodesic
manifold at 0, that could be supposed to be defined by the equations
*P+1 « ... -J* - 0
We will have then
V "ίίί1λ + ***" + Y22 + - + V]<fal ^ ■" ** '
the Integral being extended 1n the normal Euclidean space to a hypersphere of
radius R and centre 0.
It will only be necessary to consider the squared terms 1n γ.,. On the
other hand, we have
LV dr1 ax2 ... *P - J fr2 αχλ ax2 ... at ■
Let
V -at
о
be the expression /or the volume of an Euclidean hypersphere of radius r,
where α 1s a conveniently chosen constant. Its surface 1s
We have
fr2 άχλ ... at - fr2 Sc(r)dr - ρα/V*1 dr
"F^^-F^V2

Normal Riemarmian Coordinates
253
The sum of the coefficients of the squared terms in γ,Ί + γ„ + ... + γ is
I I СС ПП
Ι Σ R.... - - Ι К ,
3 (iJ) w 3
where К denotes the curvature at 0 in the direction of the manifold V .
Ρ
He thus have
V я Vo ' 3(p + 2) K VoR
from which
V -V „
о _ К
7' W*b (37)
V R
о
Finally, the relation
v - <* - w+ijfaRP+2
yields on differentiating:
S « poRP"1 - 1 KoRP+1 - $o (1 -■£ R2) .
*
from which
S -S v
о К
S R
о
?B¥
The general results established in this section were first given by H. Vermeil
(Giitt. Nachr. 1917, pp. 334-344).

Chapter XI
SYMMETRY AND PARALLEL TRANSPORT, SYMMETRIC SPACES
I. SYMMETRY AND PARALLEL TRANSPORT
236. We shall consider,in the neighbourhood of a point 0, the point
transformation which takes a point Μ to the point M' symmetric to Μ with
respect to 0 on the geodesic MO produced from 0 1n the opposite
direction. This 1s another way of saying that the length OH1 of the arc 1s equal
to 0M. This transformation, that we refer to as symmetry with respect to 0,
1s not 1n general an 1sometry. To every vector stemming from M, there
corresponds exactly a definite vector from the symmetric point M'. If the first
were to represent the velocity of a point 1n motion, the second would be that
of a moving point constantly symmetric to that of the first.
In terms of normal co-ordinates x1 attached to 0, the symmetry of the
points 1s related by equations
(x J - -x ;
for the vectors with respect to the natural frames associated with the normal
co-ordinates, 1t 1s related by the equations
(XV - -Χΐ
Let us consider two points Μ and M' both symmetric with respect to 0 and
both very close to 0, as well as two symmetric vectors x, x' stemming from
Μ and M1 respectively. The vector x'\ attached to M1 and equal and
opposite to x1, has the same components as the vector x; consequently the
quantities Гл. being zero at 0, result from the transport by parallelism
of the vector χ from Μ to H\ We thus have the new construction
following the transport by parallelism.
To transport by parallelism a vector x, with its point of attachment
going from Η to a point M' infinitesimally neart it suffices to oonetruat
the symmetric veotor x' of x, with reepeot to the centre of the geodesic
aro MM'л then the veotor x" ie equal and opposite to x\
This construction 1s called transport by symmetry.
237. The above construction naturally leads to a result that 1s exact up
to Infinitesimals of greater than the first order where the principal
Infinitesimal quantity 1s MM1. He are going to show that if the transport by
parallelism is made along the length of the geodesic aro MM1, the transport
by symmetry gives rise to a result that is exact up to third order infinitesi~
mals.
255

256
Syrmetria Spaoee
Let us note that 1n this case» 1f rectangular frames (R) are used» these
being adapted to the normal co-ordinates 1n question (see no. 216), the parallel
transport along a geodesic passing throunh 0 does not alter the components of
this vector with respect to the frames R. Now the components are none other
than the forms ыг{хлах)> where numerical values are given to the differential
<£гъ that have the components X1 of the vector taken with respect to the
natural frame having the same attachment as that associated with the normal
co-ordinates. This being the case, to transport by symmetry at 0 amounts to
taking the quantities шъ(х,<£г) to the quantities ωτ(-χ,<£τ). The parallel
transport along the geodesic MM1 preserving the numerical values of the forms
ω1 brings us to comparing ω {x,dx) and юг(-х,<£г). Now the limited
expansion obtained 1n Chapter X, no. 225, relation (19) gives
J{g,dg) - J{-g,dx) - I RrfWe xrxkx*dxh . (1)
which with ax being finite, 1s Indeed a third order Infinitesimal quantity.
238. Let us determine those conditions that must be satlslfed 1n the
given R1emann1an space so that the transport by symmetry gives a construction
for parallel transport along the geodesic which 1s exact to fourth order
Infinitesimals. We have to state that the cubic forms
%■ ■ W.л**' <2>
are Identically zero. This relates to a certain number of linear relations
with constant coefficients between the components of the derived Rlemann-
Chrlstoffel tensor. It 1s clear that once given a geometric Interpretation of
these relations, they are Independent of the choice of frames. We are going
to utilise this remark under the same conditions as for the B1anch1 Identities
of no. 191 ■ relation (13). Ue note 1n this case that the components R--£wg
transform under a change 1n frames as the products Χ.Υ.Ζ,Τ,υ. of the
components of five arbitrary vectors. He shall consider then a simple
Infinitesimal rotation of the frame parallel to the biplane determined by the basis
vector e and β .
г в
If ε 1s the Infinitesimal angle of rotation, the component X of a
2*
vector undergoes an elementary enlargement εΧ , and for the component X ,
в в
1t 1s -εΧ » the other components being unchanged. This results (see no. 212)
1n such an elargement for the component R—ll/, and will be a factor of ε.
the sum of the components obtained by replacing wherever 1t occurs, the Index
r by β, minus the sum of the components when 1n a similar way « 1s
replaced by r. Ue shall denote the rotation 1n question by {re}.

Syrm&trio Spaoee 257
239. Let us apply the above considerations to the coefficients of (χ1)
1n the form F,,; on account of this coefficient being zero» we shall have*
R..;.» - 0 . (3)
The rotation \Jk) will permit us to deduce this following 1t. viz
"W - ° <4>
and sinllarly. the rotation iik) applied to (3) will give
2Wi + W*"° · (5)
The B1anch1 Identity
ij'ij/k ijjk/г ijik/j
permits writing (5) 1n the form
from which, by changing the two Indices г and j\
R..,. .. - 3R.... .. ■ 0
гззк/г bjik/j
From these last two equations, together with (5), we obtain the equations
R...... » 0 , R....« * 0 , R.j..,. ■ 0 . (6)
tjtfc/j гагз/к i^jfc/г
The rotation {jt) applied to the first equation (6), eventually yields
R + R я η
from which, by a cyclic permutation on the Indices j, k, U results 1n two
new relations which show that the sum of any two of the three components
■Wi' Rifc£l/j· \Щ/* 1s zer0- They are therefore a11 zer0·
R...,,/„ ■ 0 (7)
By applying the B1anch1 Identities»
*
In equation (3) and those following, 1t 1s not necessary to sum with respect
to Indices that occur several times, these only having Individual values. In
all of these equations, the Indices denoted by different letters are assumed
different.

258
Symmetric Spaces
Rijik/t*Rijkl/i ' hjil/к " ° '
the new equation is deduced from (7), viz
The equations (3) to (8) prove that the only components that could be nonzero
are those with five distinct Indices. Now the rotation {im) applied to (8)
gives the new equation
kunj/l· kttj/m
from which
Ktij/m Kljm/i " «furi/j '
but as the sum of the last three components 1s zero, and whatever the Indices
it j, k, I, m, 1t 1s
R-t../ я ° · (9)
All of the components of the derived tensor of the R1emann-Chr1stoffel tensor
are therefore zero.
Theorem. In order for transport by symmetry to determine parallel transport
along an infinitesimal geodesic aro (to infinitesimals greater thai the third
order), it ie neaeeeary and sufficient that the derived tensor of the Riemann-
Chrietoffel teneor should be zero.
We note that since the derived tensor of the tensor of R1enann1an
curvature 1s zero, the geometric relevance 1s that parallel transport preserves
Riemannian curvature.
The proof given shows that this property 1s a consequence of the equations
1n (3) alone; these state that the Riemannian curvature of the space in the
direction of a planar element has its absolute derivative zero in any dweotton
of this planar element.
240. We now Intend proving the following theorem:
Theorem. If the derived teneor of the Riemann-Christoffsl tensor ie zero,
the transport by symmetry rigorously determines the parallel transport along
a geodesic arc, the necessary and sufficient condition for sytmetry with respect
to a point to be an isometry is that this tensor is zero.
Effectively with parallel transport preserving Riemannian curvature, the
components R-му. of the R1emann-Chr1stoffel tensor, with respect to the

Syrmstric Spaoee
259
system of rectangular frames R adapted to a normal co-ordinate system
attached to 0, are absolute oonatante. Let us then take the fundamental
differential equations which define the forms ω1 {tta;da) and ω., {t,a;da)
[Chapter X, no. 217. equation (7)]:
3ω* . i . k-
ΐ- Rtf» aW
(ΊΟ)
Ие are going to deduce the relations
u [t,a;da) я ω {tt-a;da) (11)
Effectively we put
ω {tta;da) - ω [tt-a;da) =* ω
u..(t,a;da) + ш^,{Ь,-а;аа) "ω.. .
A straightforward calculation yields
This 1s a system of homogeneous linear differential equations with constant
coefficients. As the Initial values of the unknown functions are zero for
t-0, all these functions are zero; from this results relation (11).
Now on replacing t by 1, a by χ and da by dx 1n this
relation, we obtain the Identities
ω ж [x;dx) я ω {-χΛάχ) (12)
They show that:
1° The transport by symmetry along a geodesic arc. rigorously determines
the parallel transport along this arc.
2° The symmetry with respect to 0 leaves the fundamental form unchanged.
I.e. 1t 1s an Isometry.
241. It now remains for us to prove that 1f the symmetry with respect to
a point 1s an Isometry, the parallel transport preserves R1emenn1an curvature.
This 1s clearly geometric. He start from a planar element attached to M,
defined by the Ыvector formed by the two vectors χ and y. The symmetry
with respect to 0 transforms these vectors Into two vectors x* and y*

260
SyRwnetrio Spaoee
attached to H1. As the symmetry 1s assumed to be Isometric 1t preserves
R1eaann1an curvature, this being uniquely determined by the fundamental form.
The R1emann1an curvature along the planar element [xy] 1s the same along the
planar element [xVJ; but these two planar elements are deduced from each
other by parallel transport up to third order Infinitesimals. Parallel
transport thus preserves R1eaann1an curvature up to third order Infinitesimals and
consequently the absolute derivative of the R1eaann1an curvature 1s zero and
thus 1t 1s rigorously determined. Q.E.D.
Thus the theorem 1n no. 240 1s proved completely.
II. SYMMETRIC RIEMANNIAN SPACES
242. R1emann1an spaces which we call eyrmetrio can be defined by two
properties that we are going to prove to be equivalent:
1° Symmetry with respect to an arbitrary point 1s an Isometry.
2 Parallel transport preserves R1emann1an curvature.
He shall show that the determination of symmetric spaces reduces to an
algebraic problem.
Let us Introduce at a point 0 1n space normal co-ordinates
corresponding to a rectangular frame (R ) attached to 0. The components R··^ of
the R1emann-Chr1stoffel tensor with respect to the frame adapted to this
co-ordinate system are absolute constants that satisfy the classical symmetry
relations
ijkh " Jikh ijhk
(13)
гзкл vkhj ihjk
On the other hand, by taking the absolute differential of each of the components
as zero, we obtain the relations
R .,,ω. + R. , -ω. + R.. ,ω, + R... ш, * 0 ,
rjkh гг vrkhjr tjrft kr zjkr hr
from which, assuming the Introduction of superabundant co-orda1ntes t and a
and noting that ω., -ω.., we have on applying the fundamental differential
equations 1n (10),
rjkh ггЬп ггкк jrim tjWi krbn ijkr nrSm
ti>j>kthtl,m " l»2,...,n)
He are going to prove that conversely, to every oo-ordvnate eyetem eattafying

Syrmetrio Spaoee
261
the quadratic relatione in (13) and (14). there corresponds a eymetrio Rieman-
nian враое, and thie epaoe ie locally ieometrio onto itself in an infinity
of uays.
243. Let us Integrate the fundamental differential equations
. i , k—
da + α ω. .
> * ■ К..,, α ω
(10)
with the Initial conditions ω * 0, ω.. ■ 0. This 1s a system of linear
differential equations with constant coefficients, providing for the forms
—t — 1 2 и
ω » ω.., linear expressions 1n da ,da ,.,.,άα whose coefficients are
lJ 1 2 и i
entire functions of t,a ta: ,.,.,α: . If we set t ■ 1 and replace α by
x1, we obtain the forms шг[х,ах), ω..(χ.4τ). The determinant of the coeffl-
12 и 1 2*^ и
dents of dx\ dx ,.,.,άχ 1η ω ,ω ,....ω 1s different from zero for small
enough values of the x1 (equal to 1 at 0).
But 1t remains to prove that the forms ω1, ω., so obtained satisfy the
structural equations
(15)
*· tj
It suffices, moreover, to prove that these relations are true for the forms
^(t.a-.da) and ω.^(ί,α;<ία), the exterior differentials and dH^j being
calculated as 1f t were a constant.
Let us take
dw ■ [ωω..] + ε
(16)
<*»i E GWfc] + Ϊ Кг^^Л] + Etj
where ε1 and ε., denote the exterior quadratic differential forms 1n
1 2 и tJ
da\da t...tda , which are clearly zero for t * 0, since for t я 0 the
forms ω1 and ω., are themselves Identically zero.
We are going to establish a system of linear, homogeneous differential
equations for which the forms ε1 and ε., must satisfy. Once this has been
done 1t becomes evident that these forms, which are zero for t - 0, are
Identically zero, thus proving the relations 1n (13).
We arrive at the required system of differential equations on
differentiating the equations In (16) with respect to t. It suffices 1n this case

262
Symmtric Spaoee
to note that 1f ω 1s a differential form whose coefficients depend on a
parameter t, the operations of differentiation with respect to t and the
exterior differentiation are permutable. Putting 1t another way, we have*
He apply this relation to the form ω*, by taking into account of the relatione
1n (Ί3)
Calculating d -$- and ^ №у) then gives
Comparing these two equations, gives
But the form
Rtf*Afc + 4jfcAfc + Ri*M?V* + RirfM&fc (19)
as 1t 1s zero for t ■ 0, admits with respect to t a derivative equal to
α ω (Кдо\*га + RkjhtRi1cra + Ri*MRjfc» + '^/fcJl'We* '
*Th1s relation 1s easy to prove. If for example ω- i a.,{utt)[du dJ] we
have tf |u - i —tt [dJdJdub.

Syrmetrio Spaces
263
and 1s consequently zero by virtue of the quadratic relations 1n (14) where
It Is sufficient to change к to h, h to 1· r to fc, % to r and
η to β.
He therefore have
The equations in (18) and (20) are on all accounts linear and homogeneous;
this proves that the structural equations 1n (15) are true. The resulting
space 1s on the other hand symmetric, since the form 1n (19) being zero proves
that the absolute differential of the component R--fcl of the Rlemann-
Chrlstoffel tensor 1s zero.
244. Remark. The equations 1n (14) state that a certain tensor with 6
Indices 1s zero» I.e. the tensor whose соvariant components with respect to
an arbitrary Cartesian frame are
HijkhSm Ш RrjkhRi tm + RirkhRj tm + RijrhRk tm + RiJkrRh Ы '
A necessary condition for a R1eaann1an space to be symmetric 1s that the
preceding tensor should be zero. But this 1s not a sufficient condition» as
shown 1n the example of 2-d1mens1ona1 R1emann1an spaces for which the tensor
1s always zero. In practice this states that the twice derived Rlemann-
Chrlstoffel tensor has Its components R--Wl/J_ symmetric with respect to
the last two Indices.
III. RIGID DISPLACEMENTS IN A SYMMETRIC SPACE
245. Spaces with constant R1emann1an curvature are evidently symmetric
spaces and they are the most simple. This reasoning results 1n a certain number
of properties that symmetric spaces have 1n common with spaces of constant
curvature. These properties evolve quite simply from the fact that the
fundamental form as a function of normal co-ordinates relative to the point 0 1s
Identically determined every time a rectangular frame (R*) 1s taken with
respect to which the components R-^jy. of the R1emann-Chr1stoffe1 tensor
retain the same constant values. This could be arrived at 1n two ways:
1° By replacing the rectangular frame (R ) by an arbitrary frame which
1s deduced by parallel transport along a geodesic stemming from 0.
2° By keeping the origin 0 of the frame (R ) but producing a suitable
rotation about 0 at this frame.
Now the Infinitesimal variation of R.... under the effect of an

264
Syimetric Spaoea
Infinitesimal rotation of an angle α parallel to the biplane [ее] has
Г β
been evaluated In no. 238. It results In the Infinitesimal variation undergone
by the effect of an infinitesimal rotation defined by the Ыvector with
components a..; this Is
R -ufln * R· vua- + R·· Λ + R-4 a, (2Ί)
rjkh lr irkh jr ijrn hr bjkr hr x '
Consequently, the most general Infinitesimal rotation (a.*) that could be
made about the frame R without changing the components of the Rlemannlan
curvature Is given by the system of equations
R ...a. + R. ,,α. + R.. .a, + R... a, » 0 . (22)
rjkh гг ггкл jr vjrh kr ijkr hr x '
If the number of Independent equations Is "("2~ ^ - ρ for the *^~ Ί)'
unknowns a.. ■ -a.., then the symmetric space admits a connected group of
rigid rotations about 0 depending on ρ parameters. This group Is called
the ieotropy group at 0.
From this results rectangular frames susceptible to being chosen as
original frames for a system of normal co-ordinates providing the sane fundamental
form depending on n + p parameters. Consequently the space admits a connected
group with n+p parameters of rigid displacements. If the space has constant
curvature, there Is no equation (22), the number ρ Is equal to "*"»" ' and
the space possesses complete freedom of movement (free mobility).
246. Amongst the rigid displacements of a symmetric space, let us
Indicate those that result In two successive symmetries with respect to two points
A and B; these are the transveations of E. Cartan. In such a displacement
each point of the geodesic AB undergoes along this geodesic a displacement
equal to twice the distance AB. The geodesic AB Is the basic geodesic of
the transvectlon. In the Euclidean case, the transvectlons are none other
than translations; they a eta it an Infinity of basic geodesies all parallel to
each other.
In spaces with non-zero constant curvature, a transversal only admits a
basic geodesic.
IV. IRREDUCIBLE SYMMETRIC SPACES
247. The concept of an Irreducible symmetric space relates to the Idea
of the topological product of two spaces. Given two spaces E, and E« with
respective dimensions n, and п., the topological product Ε of these two
spaces Is an (л, tri-i-dlmenslonal space, each point of which Is defined as an
ordered pair (Μ,,Μ«) of a point M, of E, and a point M- of E«.

Symmetric Spaces
265
If E, and Ep are Rlenannlan spaces, Ε Is by definition a Rlemannlan
space whose fundamental form Is the sum of fundamental forms of E, and E2-
Putting 1t another way, If we consider two points (M, ,Mj, (Mj.Mi) of Ε
that are Infinitesimal!у close to each other, then the square of the distance
between these two points Is, by definition, the sum of the square of the
distance between the two points H, M' of E, and the square of the distance
Μ«· ΜΙ of. E«.
If a co-ordinate system u1 (i * 1,2,...,n) In E. Is taken and
likewise a system v° (a - 1,2 n) In E2· then every point of Ε will be
defined by the (л,+л-) co-ordinates м1, u°. The point M, of E, could
be called the projection of (M, ,M_) of Ε onto E, and the point M_ of
E2 Its projection onto E«. If the fundamental forms of E, and E« are
respectively, then the fundamental form of Ε Is the sum
g^MdJ'dJ + y^Wv^dv* .
248. If the osculating Euclidean spaces are considered at the points
H| find ^ of E1 and E2 respectively, the Euclidean space being the
topological product of these two spaces osculates Ε at the point (Μ,,Μρ)
of which H. and H. are the projections on E, and E«. This results In
the absolute differential of a variable vector of E, admitting as
projections on E. or pn E2, the absolute differential of the projection of the
vector. If along a curve (C) of E, the absolute differential of a unit
tangent vector Is zero, then (C) Is a geodesic. Consequently the projection
(C,) of (C) on E, will be a geodesic of that space, similarly for (C«)
on E?. Moreover, two arcs equal to (C) project along two arcs equal to
(C|). It may be easily deduced that the symmetry with respect to a point
(0,,0p) of Ε Is an 1sometry. It will be the same In E, and E«> with
symmetry with respect to the points 0, and 0? respectively, and conversely.
Theorem. In order for the topological product of two Riemarmian spaces to
be eytanetria, it is neoeeeary and sufficient that each of the two врааев ie
eyntnetrxc.
249. Definition. A symmetric apace ie eaid to be reducible (irreducible)
if it can (cannot) be considered as the topological product of two other Rie-
marvrtan враавв.
A space of constant nonzero curvature Is an Irreducible symmetric space.
Effectively, let us assume It to be reducible and choose in each constituent
space a system of normal co-ordinates, I.e.

266 Syjmetrio Зраовв
for the first n1 co-ordinates x1 (·; ■ 1,2,...,и,)
for the second n- co-ordinates χα (α - n,+ 1, η, + 2.... ,η, + η?) .
It Is clear that the forms ω. are zero and consequently on account of (10)
the forms Ω. ; but this Is Impossible since
«^ я -Κ[ωτωα] (К г* 0)
250. Ue are now going to prove the following theorem which reduces the
determination of symmetric spaces to that of symmetric spaces with constant
Rlemannlan curvature of the second kind.
Theorem. Every irreducible eymetvio space hoe constant curvature of the
second kind.
We recall that a space with constant curvature of the second kind Is a
space whose curvature Is the sane In every direction In (n-1) dimensions (see
no. 200), or whose contracted Rlccl tensor is a multiple of the fundamental
tensor.
Let us attach to a point 0 In space a rectangular frame (R ) such that
о
the components of the Rlccl tensor are all zero save Rn.R,,,.. . ,R . By
I I cc rot
taking that the absolute differentials of the components of the tensor are
zero, we take as an example, the reasoning for R_-,
Rjo^i.· + Ru*»-, * 0
:ru *м?гг
or
(R22 - Rn)Ul2 - 0 .
The form ω,2 Is therefore zero If R,, t R07· Let us assume that the η
components R,,.R,,,...,R are not all equal to each other, for example let
I I CC tvt
R-n e Roo я ·-· = R but take the following components R Al „.i»...»R_
I I cc pp p+1 fP* I «П
as distinct from the first. On denoting by Latin letters i,jt... the first
ρ Indices and by the Greek letters α,β,..., the remaining (n-p), we
will have
ut*a я ° U ■ li....p» a ■ ρ + l,...,n)
We shall consider then the system of normal co-ordalntes determined by the
original frame (R ). The fundamental differential equations (10) show that
the components \a3(h of the Rlemann-Chrlstoffel tensor are all zero and
where the Indices к and h may be Latin or Greek. We can easily deduce that
the only components of this tensor that are nonzero are those whose Indices

Syrmetrio Spaces
267
are exclusively Latin or Greek.* But then the equations In (10) could be
divided Into two groups, the first being those equations In which only Latin
Indices occur and the second for only Greek Indices. The quadratic form
(ω1)2 + (ω2)2 + ... + (J*)2
then only contains the variables iti ,...,/ and their differentials, and
the quadratic form
(ωΡ+Ί)2+ (</+2)2 + ... ♦ (ωη)2 ,
contains the variables x and their differentials. The space In
question therefore Is not Irreducible contrary to hypothesis.
The components R-,-,*R22»·"tR being equal to each other swans that
the space has constant curvature of the second kind. Q.E.O.
Remark. The converse of the above theorem Is not true: a symmetric space
with constant curvature of the second kind Is not necessarily Irreducible as
It may be proved In the simple example of Euclidean spaces.
2S1. Ue shall not pursue the study of symmetric spaces any further.
They have been completely determined by E. Cartan;** the elliptic and
hyperbolic Hermltlan spaces of G. Fublnl and E. Study are Included. Irreducible
spaces have a nonzero scalar Rlemannlan curvature. If It Is positive, the
Rlemannlan curvature of the space Is positive everywhere or zero; If It Is
negative, the Rlemannlan curvature Is negative everywhere, or zero.
The complete theory of symmetric spaces Is closely tied to S. Lie's theory
of finite and continuous transformation groups, and In particular to those of
the simple groups.
* Effectively, we have R., a - R. .- - R... - 0.
" г^об wjB iBtfa
**E. Cartan, Sur une closes remarquabt* d'eepaaea do Riemarm (Bull. Ser. Math.
France, t. 54, 1926, pp. 214-264; t. 55, 1917, pp. 114-134); Sur aertain*o
riemamiermee remarquabtee dee geometriee a groupe fundamental ещр1е (Am. EC.
Norm., t. 44, 1927, pp. 345-467); Groupee βνψΐββ aloe et onverte et geometric
riemannierme (J. Math, pures appl., t. 8, 1929, pp. 1-33); La thscrie doe
groupee finie et continue et I'Analyeie situs (Memorial Sc. Math., t. XLII,
1930); Lee eepaoee riemanniene symetriquee (Verh. Int. Math. Kongresses,
Zurlck, I, 1932, pp. 157-165); Sur lee dominoes bornee homogenee de I'eepaoe
de η variables oonplexee (Abh. Hath. Seminar Hamburg, t. 11, 1935, pp. 116-
161).

Chapter XII
RIGIO DISPLACEMENT GROUPS IN A RIEHANNIAN SPACE
I. GENERALITIES
252. We have already discussed the displacements In a space with constant
curvature and In a symmetric space. Let us recall that a rigid displacement or
1sometry or, more simply, a dlsolacement In Rlemannlan space, Is a point
transformation that preserves the distances between two arbitrary points
or that which leaves the fundamental form of the space Invariant. A
displacement preserves the arc length of any curve, the area of a region on
a surface, etc. He also know that It preserves the Rlemannlan curvature at a
point along a given planar direction, etc.
In this chapter we shall be particularly Interested In continuous groups
of displacements. A continuous (connected) family of displacements constitutes
a group If for each displacement, It contains the Inverse displacement, and
If at the same time for two such displacements, It contains their resultant
whatever the order.*
Every continuous group of displacements depends on a certain number of
parameters (the order of the group). This order Is at most equal to n*ng ',
this last case Is only realised In a space with constant curvature. We have
previously seen (In no. 245) how the order of a larger group of displacements
In a symmetric space could be determined.
253. He Mve In mind the two principal problems on the subject of
displacement groups, they are:
Problem 1. To detetmine groups of tranefomatione suooeptible to analytioal
representation of a group of displacements in a Hemannian враое, and for each
of theseл to determine the corresponding Riemannian spaces.
Problem 2. For a given Riemannian space, to determine all ite rigid
displacement*.
We shall restrict our atteatlon In this chapter to the first problem*,
the second, which Is closely related to the theory of mappings between two
given Rlemannlan spaces, will be dealt with In the following chapter.
*Every continuous family of displacements forms a group If It Is not contained
In a larger continuous family.
269

270
Rigid Disptaaement Groups
254. The groups that we shall consider will be assumed to satisfy certain
analytic conditions which characterise those groups around which Lie constructed
his theory (Lie groups). These conditions are as follows:
The co-ordinates (u )' of the point transformed from the point (u )
by the trans formations of the group admit oontinuous partial
derivatives of the first two orders with respect to the ao-ordinatse и
к i
and to the parameters a of the group [the (u )' are twlce-
contlnuously differentiate functions of the u1, and the a ].
In what follows, we shall restrict our attention to only those change
of co-ordinates for which the new co-ordinates are twlce-contlnuously
differentiate functions of the original co-ordinates.
The components g. . of the fundamental tensor are assumed to be also
twlce-contlnuously dlfferentlable (<r). Equlvalently, It will coae about
that the Chrlstoffel symbols and the components of the Rlemannlan curvature
admit continuous partial derivatives to a certain order; these In turn Involve
hypotheses for the g... We shall then assume. In principle, no question of
existence of the derivatives that arise In our calculations.*
II. TRANSITIVE ANO INTRANSITIVE GROUPS, TRAJECTORIES
255. A group G of rigid displacements Is said to be transitive If there
exists at least one displacement taking any given point К to coincide with
any other given point. The group will otherwise be said to be Intransitive.
The group will be said to be singly transitive If the displacement which
takes a point Η to a point M', Is unique; otherwise It is multiply
transitive.**
He refer to the trajectory of a group G of rigid displacements as the
locus of points transformed fro* a given point by the displacements of the
group. Every trajectory ζ of the group could be regarded as evolving from
the group displacements applied to any one of Its points.
Considered as a Rlemannlan space whose fundamental form Is that Induced
by Its presence In the given space, every trajectory Σ Itself admits the
*We shall put aside the Rlemannlan spaces thesis In which the ds Is
Indefinite. We can refer to the thesis On 3-Dimensional Manifolds (1891)
by E. Cotton In which he has determined complex 3-dlmenslonal analytic spaces
admitting a group of continuous displacements.
**Th1s term must not be confused with Its usual meaning In the theory of
permutation groups on η letters where the group may be, for example, doubly traui-
tive If It contains an operation permitting a change from any two given letters
to any two other letters.

Rigid Displacement Groups
271
group G of displacements which, from this point of view, Is transitive. Let
p<n be the number of dimensions of the trajectories. If G Is of order p,
It acts on Σ as a simply transitive group. If the order Is greater than p,
It acts as a multiply transitive group. In the particular case p-1 where
the trajectories are lines, G necessarily has one, for the maximum
connected group of rigid displacements of a line Is given by the equation
u' * u + a9 where и denotes the curvilinear abscissa, calculated as
originating from a fixed point.
256. If the group G of displacements Is transitive, the given space
Is generated by the family Σ of trajectories of the group. There passes
through each point of the space one and only one trajectory, and these
trajectories depend on n-p parameters, assuming the trajectories ara p-dlmensjonal.
These (n-p) · parameters could be regarded as constituting {n-p) co-ordinates
of an arbitrary point In space. In order to specify the points In turn of a
trajectory, It Is necessary to Introduce ρ extra co-ordinates.
III. THE FRAMES ADAPTED TO A GROUP OF DISPLACEMENTS
257. He can apply the method of the moving frame by attaching to each
point In space a Cartesian frame or even a family of Cartesian frames
followlnq way: Given a trajectory Σ, we shall attach to a particular point
of Σ a particular Cartesian frame (R.) and consider all the frames derived
from (R.) by the rigid displacements of the group. This has a certain sense,
since as every point transformation of the space, at the same time that It
transforms a point Η to a point M\ transforms In a certain way, a vector
stemming from Η to one from M1*. If the transformation Is a rigid
displacement, It preserves the lengths of the vectors, the scalar product, etc.
Consequently, the transformation of a frame (F^) attached to Μ gives
a frame (Ryr) attached (Км)* attached to M1 whose basis vectors have the
same mutual scalar products as the basis vectors of (Ry): the frames (Ry)
and (Ryr) are, In other words, equal.
Every family of planes obtained by the above procedure will be said to
be adapted to the group of displacements In question. The above remarks
Immediately lead to:
Theorem 1. The aorrponente д.. of the fundamental tensor with respect to a
family of frames adapted to a group G of displacements of the враое, are
invariant under this group. If this group Is transitive, they are absolute
The components of a vector with respect to the natural frames associated with
chosen co-ordinates, transform as the differentials of the co-ordinates by the
effect of the point transformation In question.

272
Rigid Displacement Groups
components; If It Is Intransitive, they only depend on the (n-p) parameters
that specify the group trajectories.
258. There Is another similar theorem ralatlng to the forms ω*, иА
which define the position ralatlve to two frames that ara Infinitesimally near.
Let (Ry) and (Ry,) be two such frames of the family attached to two
Infinitesimally near points M, M\ A displacement of the group transforms
them Into two other frames (O, (R..,) both Inflnlteslmally near. The
configuration determined by (IL.) and (FL.,) Is seen to be equivalent to that
formed by (Ry) and (Ry,);* we thus have
Theorem 2. The forme ω л ьг. that define the structure of the space, with
respeot to a family of frames adapted to a group G of displaoements of the
spaoea are invariant under this group.
To complete matters, we Include a theorem that relates to the case whera
the trajectories ara transformed In a simply transitive way by the qroup G.
To each point of the space Is attached a single frame, and the forms u£ ara
i
the linear combinations determined by the forms ω
ufi - γ.Vе ■ 0)
г Ч к * '
Theorem 3. The coefficients γ. . (the generalised Ckrietoffel symbols) are
invariant under the group of displaoements if the order of this group is equal
to the number of dimensions of the group trajectories.
This theorem Is In fact an Immediate consequence of the relations In (1)
[see p. 271] and the Invarlance of the forms ω and J..
The same theorem applies to the γ... as well as to the components of the
Rlemannlan curvature tensor (the Rlemann-Chrlstoffel tensor).
In certain problems, It would be convenient to employ rectangular frames,
as Indeed It Is In other cases where It Is praferable to have a wide range of
choice of frames to be adapted to the group.
The components шг with respect to the frame (R^) of the Infinitesimal
vector Wf are equal to the components (шг)' with respect to the frame
(Ru) of the Infinitesimal vector ΗΝ*". Moreover, the quantities trf7. which
π ι
determine the components of the basis vectors of (Ry.) with respect to
(Ry), are equal to the terms (ui)\

Rigid Displacement Croupe
273
IV. RIEMANNIAN SPACES ADMITTING A SIMPLY TRANSITIVE GROUP OF DISPLACEMENTS
259. On account of the relations derived In the preceding sections, we
can Imagine the determination of the Rlemannlan spaces admitting a group of
displacements as one that amounts essentially to determining those spaces
which are transformed transitively by this group. It Is from this case that
we shall commence by firstly assuming that the group Is simply transitive.
Let us take a system of frames adapted to the group G, with forms
шг and hi Invariant under the group· of which the first η are linearly
Independent. We are going to determine the conditions that these forms must
satisfy In order that the space admits a simply transitive group of displacements
The displacements of the group G, If It exists, will be given by the
Integration of the total differential equations.
</(u\ifu·) - шг(и,аи) , (2)
where the (u1)' are unknown functions of the и1. As there must exist a
solution, such that for given values of the u\ there corresponds a given
value of the (иг)', the system Is completely Integrable.
It Is possible to find the necessary conditions directly In order for
this to be the case. Let us form the exterior differentials <&>г of the forms
г 1 2 η
ω that we shall express as exterior quadratic forms In ω ,ω ,...,ω :
** - ϊ ·»[ΆΛ («^-V) (з)
The group G leaves the forms шг Invariant and consequently their exterior
differentials ашг, and will leave Invariant the coefficients с,.г. As G
Is transitive, these coefficients are therefore constants.
Theorem 1. The forme ω relative to a system of framee adapted to a simply
transitive group of displacements eatiefy the relatione in (3) where the
coefficients a,, are aonetante.
260. Theorem 1 gives rise to Important theorem which Is complementary,
and which we Intend proving.
Theorem 2. Every sinply transitive group in η variables oould be regarded
ae a simply transitive group of dieplaoemente for an infinity of n-dimeneional
Riemannian spaces. *
♦Concerning simply transitive groups, this theorem solves the first part of
problem 1, or rather, amounts to the study of the simply transitive groups
that arises uniquely from the theory of groups.

274
Rigid Displacement Qroisps
Let
(и*)1 - /(κ,α) U * 1,2,... ,π) (4)
be the finite equations of the transformations of the group, where we assume
that the f are twlce-contlnuously differentiate functions of the arguments
19 19 V
и ,u ,...,un,a ,α ,.,.,α". Ue regard the и as the co-ordinates of a point
In n-dlmenslonal space. Let 0 be a fixed point with co-ordinates (иг)в,
and let Η be an arbitrary point with co-ordinates и.
There exists a transformation Τ of the group taking the point И to
ka
the point 0, and the parameters a of these transformation are obtained by
solving the equations In (4) where the left hand sides are replaced by the
constants (иг)о. Now let M* be a point with co-ordinates Ыг)' and let
T. be the transformation taking M* to 0 with parameters b that are
obtained by similar means. It Is possible to go from И to M1 by firstly
applying the transformation Τ which takes Η to 0, and then the
transformation Τ "Ί, the inverse of Т., that takes 0 to M*. Let Τ - Τ "ΊΤ
ί ΐ aba
be the resulting transformation. Now let (u + du ) be a point M,
Infinitesimal 1 у near to Η and let. M', the point obtained from M, under Τ ,
be Inflnlteslmally near to M\ Now let 0, be Inflnlteslmally near to 0
with co-ordinates (иг) + Шг) , that Is deduced from ^ by the
transformation Τ ; It will be deduced from M* by the transformation T,. Ue will
a I и
then have the relations
{aVi) . »/(«,«} . VV.*) d{uky .
° Ъик Э(и*)'
Let us substitute the parameters αι ,α2,..·.,αη In the derivatives V fcW
к *"
by their values as functions of the и , and do the same for the derivatives
Э^К'Ь) . Ue will then obtain relations of the form
ω [uidu) * ω (u'idu*)
from which the lemma that follows, Is a fundamental theorem 1л the theory of
groups.
Lemma. Every ounply transitive group oan be defined ав a set of
transformations that leave invariant the η linearly independent differential forma
ω (u',du).*
*The transformations that leave invariant the forms </ can at most
depend on η parameters, for the equations In (5) can only admit a solution
that at most makes the given values of the (u1)' correspond to the given
values of the zA See Note V.

Rigid Dtaplaoment Groups 275
The forms шг are linearly Independent since the determinant of the coef-
flclents of the du In these forms Is the functional determinant of the
i к
functions f with respect to the variables и , and this determinant Is
equal to 1 for the Identity transformation of the group.
261. The above lemma proves Theorem 2. In fact the group In question
leaves the forms ω1 Invariant as well as all the quadratic differential forms
ff.
У
with constant coefficients. On the other hand, the form g.j^uf defines an
n-dlmenslonal Rlemannlan space under the sole condition that the coefficients
д., are chosen so as to make the form definite positive.
It Is Important to note that If we change the choice of the constant values
of the ff.jt the Rlemannlan spaces obtained are. In general, no longer
Isometric to each other.* The forms ω are then exact differentials that
could be taken to be du , and the corresponding differential forms have
constant coefficients. All of the resulting spaces are locally Euclidean and the
group G Is then the translation group.
262. The constants c* are called the structure constants of the
group 6. They In fact define the structure, or the law of composition, of
the group transformations In the sense that If two simply transitive groups
with the same structure constants are considered, then they are similar,
I.e., they can be regarded as providing the same geometric transformations
In a space where there Is a co-ordinate change. Now let ^(ν,άυ)
be η linearly Independent differential forms satisfying the same relations
as In (3).
The equations
ufivido) - шг(и;аи) (г - 1,2,..,,η) , (6)
where the variables иг are regarded as unknown functions of the variables
и , constitute a completely Integrable system. We have, In effect, on setting
^ - ω1 - ei, the Identities
,йг 1 г г к h к h-\ m 1 irak~h ah k-,
Ζ ckh '■ωω - ω ω J Τ сьу, Le ω - θ ω J
*There Is however a rather special case whece one and the same Rlemannlan space
Is obtained; It Is where the constants аъЛ are zero.

276
Rigid Displacement Groups
or
df = 0 (mod θ\θ2 θη) ,
that express the complete Integrablllty of the system (see Note V).
Every solution of the equations In (6) could be regarded as defining a
co-ordinate change In the space of the u\ and by this change of co-ordinates
the set of transformations which fix the forms шг becomes the set of
transformations that fix the forms ω1. We thus have the same group operating In the
same space, but with two different analytical representations.
263. We add that every Rlemannian space admitting a simply transitive
group G could be regarded as the representative βραοβ of the group. Its
sense Is that every transformation of the group can be represented by a point
Μ to which the transformation takes a point attached to 0, chosen once and
for all. The product of the transformation represented by the point Μ and
the transformation represented by the point A (the first named
transformation being performed first) Is represented by the point M* to which Μ Is
taken by the displacement taking 0 to A; this can be symbolically expressed
by the relation
TM· = tatm ·
If A Is regarded as a fixed point, then this relation simultaneously defines
an operation (a displacement) on points Μ of the space, and an operation on
the transformations T« of the group.
264. It Is evident that without changing a simply transitive group G,
It Is possible to replace the forms </ by η other forms that are linear
combinations with constant coefficients of the first. Geometrically, this
amounts to changing, In any one of the Rlemannian spaces admitting G as a
group of displacements, the frame attached to a particular point 0 to
another frame. The change in question modifies the values of the constants
ckh' Here, these constants play the part of the components of an
antisymmetric tensor with three Indices with respect to the first two Indices. It
Is this tensor considered In Its entirety that defines the group structure
and not Its Individual components alone.
We might pose the question of the existence of simply transitive groups
with arbitrary structure constants аъ^· This Is quite simple. The
exterior differentiation of the equations In (3) effectively gives
к h-, л __ m iT. к h..S-л
a„ [ώα ω ] ■ 0 or а^ °т1^ш ω J
On combining the terms In [ωωω ], we obtain the quadratic relations

Rigid Displacement Groups
277
akh°ml + °hl°mk + °tk°mh ' ° <**fc-M " 1.2,....n) (7)
These relations are classic In the theory of groups.* We shall see In a coning
section (no. 268) that conversely, If these relations are true, then there
exists a system of η linearly Independently differential forms ω (u,du)
satisfying the relations In (3).
V. CANONICAL CO-ORDINATES IN A RIEHANNIAN SPACE ADMITTING A SIMPLY TRANSITIVE
GROUP OF RIGID DISPLACEMENTS
266. The canonical co-ordinates that we are going to Introduce, are
similar to normal Rlemannlan co-ordinates, but they are nevertheless distinct.
They are closely related to the canonical parameters of Lie's theory of finite
and continuous groups.
Let us assume that attached to different points of the space, there Is a
system of frames adapted to the group. Let 0 be a point In space and let
12 и
us denote by α ,α ,.,.,α , the direction parameters of a direction from this
point with respect to the frame (RJ attached to 0.**
We shall consider the continuous family of points M, and consequently
the frames (Ry), defined by the differential equations
ω [w,du) ■ a dt ,
where t Is an Independent variable, with the Initial conditions и1 я (иъ)
i °
for t-0 [the (u ) are co-ordinates of the point 0]. The point Μ will
describe a curve (C) beginning at 0, and whose tangent at each point will
12 η
have as Its direction parameters a ,a a with respect to the frame
(Ry). The tangential line element at Μ to the curve С will then result
as the tangential line element at 0, under the displacement of the group G
that takes 0 to M. All of these displacements that take 0 to the
different points Μ of the curve (C) form a group. Let us consider, in effect,
the configuration formed by the Infinitesimally near frames (R„) and (Ry)
where Μ and M. are two points of the curve (C) with respective
parameters t and t + ε, as well as the configuration formed by the frames (RN)
and (Rj. ) where N and N, are two points of (C) with respective
parameters t* and t'+ε, ε being the same Infinitesimal. These two figures
are equivalent on account of the equivalence In each case, of the forms ω\
ιΑ relative to these two configurations. The rigid displacement that takes
*See e.g. E. Cartan, La theorie des groupes finis at continue at la Geometric
differentiate. Chapter XIII, pp. 223-241.
**The parameters a 9a ,.,.,α Introduced here are not to be confused with
the parameters of the group that were Introduced In no. 254 and In the
equations In (4).

278
Rigid Displacement Groups
Μ to N thus takes M, to N,, and eventually the point with parameter
t + h Is taken to the point with parameter t' + fc. This displacement Is
besides that which takes 0 to the point of (C) with parameters t'-t; It
allows the curve (C) to glide over Its own path. The equation that defines
the transformations of this one parameter group that fixes the forms a%dt%
I.e. the form dt. Is simply t' - t+constant. It can be seen that the total
group G admits an Infinity of one paremater subgroups, each of which
corresponds to a choice of direction parameters α . The curves (C) are then the
trajectories of these subgroups.
We might add that the curves (C) are not arbitrary; each of them has a
constant curvature. The curves (C) are developed on the Euclidean tangent
space at one of their points, describing straight lines or circumferences If
n-2, and If n-3, straight lines, circumferences and circular helices.
It Is clear that the point of the curve (C) defined by the (n+1)
super-abundant co-ordinates a ,a an,t, only In practice depends on the
12 η
η products a t,a t,...,a t, and the differential equations In (15) may be
written
ω (u'ydu) я d[a ,t)
The quantities хг - a't are the aanonioal parameters of the rigid
displacement that takes the point 0 to the point Μ with parematers t on the
12 я
curve (C) with parameters α ,α ,...,rf\ They are the aanonioal oo-ordinates
of the point Μ of the Rlemannlan space.
Every point of the space sufficiently near to 0 admits fixed canolcal
co-ordinates, but It Is not for certain that this Is true for every point of
the space* and It Is possible to give examples where, In fact, this 1s not the
case.
267. It Is possible to determine the expressions for the forms ω1 when
canonical co-ordinates are Introduced; these expressions, however, do not
depend on the point 0 chosen as the origin.
Let us Introduce new superabundant co-ordinates аг,ь. On displacement
along a curve (С) {ааг = 0), we have </ ■ α dt. Let us put
ω ■ a dt + ω {tta \da ) ; (9)
the forms u^ are zero for t-0, since If we fix t and the fixed value of
t Is zero, the frame (R ) does not vary and the forms ω are zero. Let
us recall from the equations of (3) In no. 259,
♦Alternatively, It Is not for certain that every point of the space could be
deduced from 0 by a transformation of a one parameter subgroup of the total
group.

Rigid Displacement Groups
279
ашъ ■ j ffy^l·» ω ] , (3)
where the forms шг are replaced by their expressions In (9), the terms that
contain dt. We find that
IdSdt·} + [dt f£] - Ol0fak [dtrf4 ,
from which
~5F ώ * °khaui · (Ί0)
We thus have, for the determination of the forms ω1 that are zero for t-0,
a system of ordinary differential equations with constant coefficients (the
к к —i
a and da are In fact regarded as constant parameters). The forms ω
12 η
will be entire functions of t,a 9a:,...,β . This In turn proves for us the
theorem of no. 262 In which two simply transitive groups having the same
structure constants are similar.
If In the forms ω* the variable t Is replaced by 1 and the arguments
a'
-\a ,.,.,α" by χ ,x ,...,xn, then we shall obtain the forms </{χ,άτ)
expressed In terms of the canonical co-ordinates. The coefficients of the
differentials dx In these forms are Integral functions of χ ,χ ,...,χη.
A Riemannian space admitting а simply transitive group of rigid displacements
thus becomes an analytic space ωΗβη canonical co-ordinates are introduced. In
practice we can only affirm this for a sufficiently small region V about
к
the origin 0, In which the determinant of the coefficients of the dx In
the forms юг Is nonzero. But we can show that It Is possible to cover all
of the space by overlapping neighbourhoods; In each of which It Is possible
to Introduce co-ordinates such that the fundamental form becomes analytic.
To account for how we steer through the proof, It suffices to note that If A
Is a point of the neighbourhood V , for example near to the boundary of V 9
we will only have to apply to V the rigid displacement Τ that takes 0
to A. We will obtain a certain neighbourhood If. of A, In which a new
system of co-ordinates will be Introduced. We shall assign to every point Μ
of V. a new co-ordinate that will be the canonical co-ordinate of the point
In V that the displacement Τ has taken to M.
268. We have stated In no. 265 that given a system of constants o^ ■
"ahk satisfying the quadratic relations In (7), there exists forms шг
satisfying the relations In (3). We can now prove this theorem by showing that
the forms (^(t.a-.dc) that are solutions of the equations In (10) satisfy
the equations
л г _ 1 ir к h-\
αω ■ -к л, [ω ω J

280 Rigid Displacement Groups
where t Is regarded as a constant parameter, the аг as variables and the
da as their differentials. It Is sufficient to apply the argument previously
applied for a similar theorem In the theory of symmetric spaces (Chapter XI,
no. 243). The proof Is left to the reader, It Is, however, more
straightforward than that of no. 243.
269. We can prove group theoretically the following theorem of which
the last section only accounts for a small part of It.
Theorem. Given a system of constants a., ■ -a** satisfying the quadratic
relations in (7), there exists an analytic, normal, simply oormeated Riemannian
space admitting a simply transitive group of displacements whose o,-. are the
structure constants.
By a previous remark there exists, In general, an Infinity of them, all
of which are homeomorphlc to each other. There can also exist analytic,
normal, but not simply connected, Rlemannlan spaces. These are problems
concerning the topology of Lie groups.*
VI. CANONICAL CO-OROINATES AND NORMAL CO-ORDINATES
270. There is a great deal of similarity between the canonical
co-ordinates and normal Rlemannlan co-ordinates, but In general there Is not exact
equivalence. It Is possible to find a case whereby the canonical co-ordinates
are normal co-ordinates. For this It Is evidently necessary and sufficient
that the trajectories (C) of the given one parameter subgroups are geodesies
of the space. Now on displacing along a trajectory with parameter аг, we
have
°°4· a
St °ayiK
η
In order that this trajectory Is a geodesic, It Is necessary and sufficient
that the quadratic form у.к.аъаН Is Identically zero, I.e. the coefficients
γ. , (and consequently Ύ·^,) are antisymmetric with respect to the two
extreme indices. Now we have
г г г
\ h ' yh к ' °kh
from which there results
*See E. Cartan, La theorie dee groupes finis et continue et I'Analysis situs
(Mem Sc. Math XL11, 1930). Also by the same author, La topologie dee groupes
de Lie (Exposes de Geometrle, VIII, Hermann, 1936).

Rigid Displacement Groups
281
г я 1 г 1
yk h J°kh ' ykih 1 °кЫ *
As the forms ω., are antisymmetric with respect to the two Indices» the
result Is that the соvariant components л, . of the structure tensor β*
(already antisymmetric with respect to the first two Indices) are also
antisymmetric with respect to the last two Indices, and from this It may be easily
deduced that the tensor a* Is a tr1vector. We then have In this case
4·ταΆ°" · (11)
Conversely, If the tensor β* Is a trivector, the difference γ. .. - γ... я
°kh' cnan9es 1*s s*9n DV changing the Indices к and г
On the other hand, as γ,
two Indices, we shall then have
On the other hand, as γ,., Is antisymmetric with respect to the first
or
ykih ' yhik + yikh ' yhki
ϊ-ты. + Υ
ihk ТШ u *
There Is then antisymmetry with respect to the extreme Indices, whence the
following:
Theorem 1. In order that the oanonioal co-ordinates in a space admitting a
sunply transitive group of displacements are to be normal Riemannian
coordinates, it is necessary and sufficient that the tensor a*, is a tri-
veotor.
271. The condition obtained Involves both the structure constants o^
of the space and the components д.. of the fundamental tensor with respect
to the system of frames adapted to the group. Given a group for which there
Is a Riemannian space admitting this group as one of displacements and such
that the canonical co-ordinates are simultaneously normal co-ordinates, then
It Is ncessary and sufficient to obtain the constants д., susceptible to
being coefficients of a positive definite quadratic form which satisfies the
relations
<W + S*°H - ° Ю.*-1.2 n) . (12)
This last condition puts a considerable limitation on the choice of the group.
272. We know that the Euclidean space normal to a point 0 of a
Riemannian space osculates the space at this point. It Is therefore the same for the
canonical Euclidean space If the given space admits a simply transitive group

282
Rigid Displacement Огоира
of displacements such that the canonical co-ordinates are normal co-ordinates.
The converse is true.
In effect, Integrating the equations In (10), of no. 267, easily yields a
limited expansion of the forms ω1, I.e.
ω ■ ax + j aj. χ αχ ,
from which we deduce a limited development corresponding to the fundamental
form
2
where ав denotes the fundamental form of the Euclidean space with
coordinates x1,
9.
del * g.-dx'dJ .
It can be seen that the condition for the canonical space at 0 to osculate
the Rlemannlan space Is the condition already obtained In (12) In order that
the canonical co-ordinates should be normal co-ordinates.
Theorem 2. In order that the canonical co-ordinate* of a apace admitting a
simply transitive group of displacements are to be normal co-ordinates it ie
necessary and sufficient that the canonical Euolidean space at a point
osculates the Riermmnian apace at that point.
273. We are going to Include an extra property of the spaces which are
of Interest to us.
Theorem 3. Every Riemannian space admitting a simply transitive group of
displacements, such that the canonical co-ordinates at a point are normal
coordinate, is a symmetric space.
It suffices to show that the derived tensor of the Rlemann-Chrlstoffel
tensor Is zero. Let us firstly determine the components of this tensor. On
account of (11)· we have
u-\°ika»k · <13)
from which
1 к
г j 2 гзк
On taking Into account the relations In (7) of no. 202, and the quadratic
relations In (7) of no. 265, we find that

Rigid Diaplaoentent Groups
283
from which
d = 1 m
If the absolute differential of R^w, 1s formed, ** obtain (noting that the
components RWi,L are themselves constants):
By replacing the ьР. by their values In (11) and taking Into account the rela-
tlons In (7), we ascertain that the absolute differential In question Is zero.
274. The Rlemannlan spaces admitting a simply transitive group of
displacements and whose canonical co-ordinates are simultaneously normal
constitute a particular but Important class of symmetric spaces. It Is possible to
show that the group G of displacements In question arises out of a larger
group of displacements whose order may be twice that of G.
Let us recall (see no. 263) that If the point M' results from the point
Η under the displacement Тд that takes 0 to A, we have the relation
TM, - ТДТМ . (14)
This being the case, the trajectories (C) of a one-parameter subgroup of G
are geodesies In this case, such that the displacement T„ which takes 0 to
Μ and the displacement T», which takes 0 to the symmetrical point M' of
Μ with respect to 0 are Inverses of each other. The relation
TM, - <ΤΜΓΊ (15)
thus defines the symmetry with respect to 0 and Is an 1sometry.
With this established, let us take an arbitrary point И, and In
succession, make the following Isometrles;
1° Symmetry with respect to 0, taking Μ to P.
2° A displacement Тд taking A to 0, and Ρ to Q.
3° Symmetry with respect to 0, taking Q to M'.
On account of (15), a straightforward calculation yields
VVa ■ (16)
The transformations In (16) actually define a new group of displacements G*
that Is, In general, different from G. On combining, In turn, the
displacements of G with those of G*, we obtain a new group Γ defined by the

284 Rigid Displacement Groups
relations
V ' Wb <">
This group Γ, depending on two arbitrary points A and B, Is In general,
of order 2n.
Let us note that the symmetry with respect to the point A Is defined by
V ' tatm\ ■ <18>
This relation states, In effect, that the two transformations Т. Ту, and
TA Μ are 1nverses of Μ0ίι other, I.e., that the points Ρ and P' which
are derived from N and M' by the displacements that take A to 0 are
symmetric with respect to 0.
Finally, the traneveotione (see no. 246) along a geodesic resulting from
two successive symmetries, with respect to 0 and with respect to a point A
of this geodesic, are given by the relation
V'Wa ■ <19>
It Is possible to show by an abstraction from Euclidean space. In which the
qroups G and 6* are Identified, that the group Γ Is of order In If
the space Is Irreducible; It Is then representative of a simple closed group.
VII. IS0GONAL PARALLELISM ATTACHED TO A SIMPLY TRANSITIVE GROUP OF
DISPLACEMENTS
275. Let us return to the general case of a space admitting a simply
transitive group G of displacements. We can define In this space an
absolute equipollenoe of vectors. It Is absolute In the sense that In order to
decide on the equlpollence of two vectors stemming from different points Μ
and N, It Is unnecessary to fix a path Joining the first point to the second,
as In the case of equlpollence given by Levl-Clvlta.
The absolute equlpollence In question, of the two vectors, Is defined by
the condition that these two vectors have respectively the same components
with respect to the frames having these points as origins, and are adapted to
the group G. Analytically, If canonical co-ordinates хг are utilised,
then every vector that could be regarded as Joining the original point M(x )
to a point (хг + dx1) 1nf1n1tesimally near, has for Its components with
respect to the corresponding frame (Ry) adapted to the group, ω (χ',άχ).
Two vectors are therefore equipollent If, for these two vectors, the forms
и>г(х;<2г) have the same values.

Rigid Displacement Groups
285
If a point A Infinitesimal1y near to 0 Is considered (0 being the
origin of the canonical co-ordinates),a vector (x,dr) joining two points Η
and M' that are Infinitesimal1y near will be equipollent to the vector
0/f If the displacement that takes 0 to Μ (and consequently (R&) to
(f^)) takes A to M\ This Is given by the relation
V " TMTA or ТнЛ· " TA ■ <20)
276. As the frames adapted to the group are all equal to each other,
then two equipollent vectors having the same components with respect to two
of these frames will have the same length. At the same time, the scalar
product of two vectors from the same origin Is equal to that of two vectors that
are equipollent and have another common origin. If then two vectors attached
to A intersect at a certain angle, the equipollent vectors taken through
another point Μ Intersect at the same angle. In such a situation we have
what Is known as an isogonal eauipollence.
277. We shall consider an equipollent vector field with a given vector.
The trajectories of this field form a congruence of curves. Let (C ) be a
representative of the curves (C) (as defined In no. 266) which depart from
the origin and which glide over each other under the displacements of a one-
parameter subgroup g of G. One that leaves an arbitrary point of A, Is
deduced from (C ) by the displacement T. that takes 0 to A and passes
from 0 to a point on this trajectory, by firstly taking a displacement from
g and then the displacement Т.. This latter displacement Is such that the
trajectory Is the locus of points Η defined by the relation
тн - V <21>
For proof of this. It can be seen that If Η and M' are two points
infinitesimal 1 у close to the curve so obtained, the displacement TM TM belongs to
g and could be represented by TQ1 where 0' represents a point
Infinitesimal 1y near to 0 on (C ).
All of the trajectories of the field In question can be regarded as
parallels. In this case we have an absolute and isogonal parallelism. Every
vector tangent to an arbitrary point of one of these trajectories Is
equipollent to a vector tangent to another given trajectory at any point of this
trajectory.
278. If the canonical co-ordinates of the space are normal, the space Is
symmetric and the curves (C) are geodesies. There Is then an absolute
Isogonal parallelism of the geodesies connected with the group G. But there
Is also a second one connected to the second group G* of simply transitive
displacements that applies to the space (see no. 274).

286 Rigid Displacement Οτοιφβ
We shall say that an Infinitesimal vector МЙ' Is equipollent of the
second kind to the Infinitesimal vector OA* If we have a relation analogous
to (20),
V " Vh or W1" ta ■ i20*)
Two vectors with equlpollence of the second kind to 0A are equipollent to
each other.
All of the demonstrated properties for the equlpollence connected to the
group G extend to that for G*. It Is the same for the absolute Isogonal
parallelism of geodesies. The geodesic parallel of the second kind to the
geodesic 0A Is the locus of points Μ defined by the relation
ТН-?ТД . (21*)
279- There exists between the two equlpollences, a straightforward
relationship:
Theorem. If two veatore are equipollent to the firet [eeoond] kind, their
symmetries with respect to α point of the вране ore equipollent of the second
[first] kind.
The relation In (20*) could effectively be written
v1 ■ vV ·
and this states that the vector, having as Its origin the symmetric Image of
Μ with respect to 0 and for Its extremity the symmetric Image of H1 with
respect to 0, Is equipollent of the second kind to the vector symmetric
to 0A with respect to 0. The theorem Is again true for symmetry with
respect to any point of the space.
From this It Immediately results that in canonical co-ordinates two vectors
are equipollent of the second kind If the terms и>г(-х;аг) have the sane values
for the two vectors.*
280. We shall name the translations of the first kind* the displacements
of the group G and the translations of the second kind* the dlsplacments of
the group G*. We shall consider that subgroup g of the translations of the
first kind that results In a geodesic 0A from 0 gliding across Its own
path.
*The χ are from the normal co-ordinates, but the шг are the forms relative
to the frames adapted to the group G and must not be confused with the frames
adapted to the normal oo-ordinates.

Rigid Displacement Groups
287
Applied to a point Η In the space, they will transform It to a variable
point by a geodesic parallel of the second kind to OA on account of the
relation In (20*). Applied to an arbitrary goedeslc, they will transform It Into
a family of geodesies which are all parallel of the first kind to each other.
There are similar properties for the subgroup g* of the second kind.
Theorem. Every one parameter subgroup of translations of the first [second]
kind results in the geodesies gliding over their own pathe, and all of these
geodeeioe belong to the scene congruence of geodesies parallel of the seoond
[first] kind. It transforms any member of a continuous oolleation of geodeeioe
parallel of the firet [second] kind to the given geodesic.
This theorem reveals the similarity between the translations that have
соме to be defined with the transformations of elementary geometry.
Another simple property Is that an Infinitesimal translation takes all
the points of the space through displacements of the same length. The
Infinitesimal vectors described by the points of the space are In fact all
equipollent to each other of the first or of the second kind.
281. A particularly Interesting case Is that of a three-dimensional
space with constant positive curvature (spherical space or elliptic space).
There corresponds a three parameter, simply transitive group whose structure
tensor о г Is a tr1vector. The Rlemannlan space Invariant under the
group Is In relation to rectangular frames and all the structure constants
reduce to a single <j,23 " c·
We obtain
(22)
■
а*л ж a[<ujaJ]
dL>2 ж ο[ω4ύι]
do3 ж oCujidg]
together with [relation (13) In no. 273],
ω
гз - - J л»! «31 " " 1 <*»
ω12 '-ϊ*]
°23 " " 4 с Ь^З1 °31 " ' 4 ° [и3^] Ω12 " " J ° h"*1 *
1 2
The space, therefore, has constant positive curvature equal to то. Let us
take оя2, without loss of generality. We know then that from no. 140, the
2
ds of the space could be represented as
de2 - dx2 + dx2 + dx2 + dx2 (23)
о \ с л

288
Rigid Dieplaoement Groups
on the assumption that the co-ordinates x. which are not canonical, satisfy
the equation
2 A 2 A 2 A 2,
X + X, + X*> + X, ■ 1
The equations 1n (22) and (23) are verified by putting
ωΊ = х^1 " x^o + χ2^3 " χ3ώ2
ω2 * Ά " x7^o + χ3ώΊ " х1^3 (24)
ω3 * Ά ' χ3ώ<> + χΊώ2 " х2^1 ·
as a stralqhtforward calculation shows, where the relation
x dx + χ,ώ, + χ·/&2 + χ3ί^χ3 ~ ^
Is taken Into account.
The geodesies are defined by equations of the first degree In ι, x,,
x-i and x.. Let us take for the point 0, the point χ -1, χ, « x« « x-
- 0. The symmetric image of the point (χ ,χ.,χ,,,χ-) Is the point (χ ,-x,,
-x2,-x3). In fact, these two points are aligned with the point 0, as well
as being at the same distance from 0, and the da does not change when
χ,, χ? and x- change sign. The result Is that If two equipollent vectors
of the first kind are characterised by the equality of the forms ω,, ω- and
ω., then two equipollent vectors of the second kind are characterised by the
equality of the forms
"l ' χο**Λ " χλ*Ό ' х2^3 + x3dx2
"2 = χ0ώ2 " *&o ' x3dx1 + χΊώ3
"3 * ХЛ " Х3*Ч> ' χΊώ2 + х2^1
2
whose sum of the squares Is the da of the space.
If we now look at an elliptic space, I.e. the three-dimensional
projective space with respect to homogeneous co-ordinates χ,, χ2· x3. then two
finite vectors limited by the points х.л у. are equipollent of the first
kind If for these two vectors, the PIQckerlan components
P01 +^23 · P02 + Р31 · РОЗ + p12
are the same. They are equipollent of the second kind If they are the
components

Rigid Displacement Groups
289
p01 " p23 · p02 " p3l · p03 " p12
which are also the same.
282. Let us represent a point x1 by the quarternlon
X ■ χ + x,t + Xyj + xJi
The quartenlon vector ιω, + ju>2 + too- Is none other than JkflT, on denoting
by 7 the conjugate quarternlon χ - χ,ί - x^j - xJc The quarternlon
vector ΐω, + juu + kL· Is none other than XrfX.*
The translations of the first kind are defined by the relations
X' - ΧΑ , (25)'
2 2 2 2
where A Is a unit quarternlon (a + at + at + at ■ 1). These leave the
forms ω. invariant, since we have
Х'Л' - Ш dl - XrfX ,
as by hypothesis, AA - 1
The translations of the second kind are defined by the forms
X' - AX , (26)
where A Is α unit quarternlon. Here the forms ω. are left Invariant, as
we have
7'<fX' * YK AdX - TdX .
These translations were originally studied by Clifford who was also the first
to have conceived the two absolute parallelisms which are named after him.
Remark. The existence of these absolute parallelisms does not generalise for
spaces of higher dimensions with positive constant curvature. Nevertheless,
In 7-dlmenslonal elliptic space, there exists an Infinity of absolute Isogonal
parallelisms that form two distinct families, but they are not connected to the
existence of a simply transitive group of displacements In this space.
*
Recall that the units t, j, к satisfy the following rules of multiplication
i2 ш j2 , k2 - -1, jk - -fcj - г ,
кг ■ -гк - j ij * -jk = к
**Papers (London 1882, pp. 181, 236, 378, 402). For further details see
E. Cartan, Leaone sur la Geometrie projective oomplexe (Gauthler-Vlllars, 1931),
1931), 2e partie. Chap IV, Section VI.

290
Rigid Displaoement Groip3
Together with the spaces that are representative of simple closed groups. It
Is the only Irreducible Rlanannlan space (see no. 249) admitting absolute
Isogonal parallelism.*
VIII. RIEMANNIAN SPACES ADMITTING A MULTIPLY TRANSITIVE GROUP OF DISPLACEMENTS
283. If an η-dimensional Rlanannlan space admits an r parameter
transitive group G(r>n)t the displacements that fix a point 0 give rise to an
r-n parameter group g. This Is the group of rigid rotations about 0, the
isotropy group or the stability group of 0. If a system of rectangular frames
adapted to the group are utilised, the stability group g represented
analytically by the group of orthogonal substitutions that bear the components of a
vector attached to 0 under the effect of the rotations of g. The
connected part of this group that contains the Identity rotation Is given by
Infinitesimal rotations, each of which Is defined by а Ыvector ζ... These
Mvectors are required to satisfy ЯУ ' - r linearly Independent relations
with constant coefficients
W«"° <°-1.2.···1^-'·) · (27)
The frames attached to different points of the space depend on r
parameters of which η are the co-ordinates и\и;,...,ия of the frame. The
1 2 r-n
remaining (r-n) co-ordinates ν ,v ,...,v serve to specify the frames
1 2 η
with the same origin. The forms ω,»ω0,...,ω are linear In du ,du,du ,
I С j П
since they become zero when the point (u ) Is fixed. As for the forms ω.,,
ί i г3
they are linear with respect to the du and the d\> . But the forms
A .iii.. only depend on the differentials du1, since when the point иг Is
fixed, the forms ω., defining the elementary rotation undergone by the
frame are none other than the components ξ., of a Mvector satlsfylnq the
relations in (27). There exists, therefore, "frl+U relations of the form
W4i" cort ■ (28)
between the nv£*l forms ω., ω... The coefficients C. are constants
since every displacement of g leaves Invariant the forms ω., ш.., and the
coefficients С .. There are the only linear relations between these forms.
284. It Is possible to deduce that the Infinitesimal rotations of the
stability group of 0 fixes a certain number of vectors stemming from 0.
Let us assume that there are ν of them that are linearly Independent, and
they give rise to a linear v-dlmenslonal manifold.
*E. Carta η and J. A. Schouten. On the Riemarmian Geometries admitting an
absolute parallelism (Proc. Amsterdam, t. 29, pp. 933-946).

Rigid Displacement Groups
291
The geodesies sternalng from 0 that are tangent to this manifold are
Invariant under the group g of rlald rotations and each of their points Is
also Invariant under g. These goedeslcs give rise to a geodesic manifold V
at 0. The stability group of any point Η of this manifold Is the group g
Itself; the Manifold V Is then geodesic at each of Its points, and Is
consequently a geodesic manifold.
Theorem. The geodeeioe tangent to a point with veotors that are invariant
under the stability group of this point give rise to a totally geodeeio
manifold.
These totally geodesic manifolds depend on n-v parameters since one,
and only one of them, passes through an arbitrary point of the space, and they
are transformed between themselves under the group of displacements G. It
could happen besides that the group g does not leave any vector Invariant;
the manifolds V then reduce to a point.
υ
285. The constants С , occurlng in the right hand sides of the
relations In (28) are not arbitrary. Let (It.) and (It.,) be two frames Inflnl-
teslmally near to each other In the system of frames adapted to the group.
We rotate each of them by an Infinitesimal amount about the origin. These
two roatlons being defined relative to (Ry) and (Ry,) respectively, by
а Ыvector with the same components ς... Let (L·) and (TL.) be the
resulting frames. The configuration formed by (Ry<) and (Ry.) Is
equivalent to that formed by (Ry) and (&.); It Is therefore the same
Infinitesimal geometric displacement that simultaneously takes (Ry) to (Ry,) and
(1L) to (1L,). This displacement Is analytically represented with respect
to (Нц) by the family of vectors ω. and Ыvectors ω·,. By changing the
system of reference that takes (Ry) to (TL), the components ω., ω..
undergo the variations
^t " ζΐΑ· ^ij ' etfrttf + 4^ik ;
with the relations In (28) being preserved, we shall have
Vf^ifc0^ + *ΐΛ*> " Zah*-hk"k ' (29)
The relatione in (29) are the ooneequenoes of those in (28) each tvne the ζ-,
eatiefy the relatione in (27). These conditions limit the choice of the
constants С .. Indeed they would automatically come about by the application of
the general method which will be outlined In the following section.

292
Rigid Displacement Groups
286. He can express the ω., as linear combinations with constant
coefficients of the ω., and of [r-n) forms ω., that we shall denote by ω,»
t t-j ι
2 r-n
The exterior differentials άω1 are the known exterior quadratic forms
of these r forms In which the ω occur In the first degree or more. On
making an exterior differentiation of the equations as written, we shall have
the relations between the C^ In question In the last section, and also the
conditions that are to be satisfied by the forms <L· . By stating them. In
the most general way possible, as quadratic forms with constant coefficients
of the ω, and the ω . On making an exterior differentiation of the equations
so obtained, It Is possible to obtain new conditions that must be satisfied
by all the constants successively Introduced. This having been done, we shall
derive a system of r forms θ ,θ2 6r (вг - ω. for i*n, βη+α β ω
for n+1 £ о £ r) which will satisfy the relations
dQi « \ C^Ve*] . (30)
with constant coefficients.
287. The coefficients Cj. necessarily satisfy the equations In (7)
In no. 265, since the exterior differentiation of the equations In (30), b.v
hypothesis, gives relations which are all consequences of the relations (30)
themselves. Thus there exists a simply transitive group g of order r
admitting the C,,1 as structure constants. Ue are going to show that It Is
possible to find η functions хг that are Independent of the r variables
transformed between themselves by G, and, moreover, that there exists an
Infinity of quadratic differential forms constructed by the хг and the dx1
which are Invariant under G.
For this we shall use the fact that amongst the equations In (30) are
found the equations
<Ч- ж Kv] ·
where the пц~ ι forms ω., are antisymmetric and the ω. are linearly
Independent. Ue shall first of all prove the following theorem that Is
Important In Its own right.
*More generally and In certain cases, more conveniently, It Is possible to
express the n\n* ' forms u.. as linear combinations of (r-n) foros ω
and ω,. The condition being that the linear forms ω are taken with
respect to the ω,, and the ω. such that on limiting the terms In ω..,
V * *J nfn-H
they form with the left hand sides of the equations In (28), a system of "^ *'
Independent forms.

Rigid Displacement Croupe
293
Theorem. Given η linearly independent differential forme ω. constructed
' 12 2·
with r>n variables и ли л...,и and their differentiate, and which aleo
satisfy .the relatione
*i" [Vw]
(where the ω.. = -ω., are equally constructed with the вате variables and
their differentiate) t then it is possible to find η independent functione
TO TO 0 0
χ |X j...jXn 0/ u jU (...(иг вштй tfezt the quadratic form (ω,) + (uu)
+ ... + (ω ) enZj/ depends on the functions хг and their differentiate.
In the first case» the equations ω, - uu - ... - ω "Ο form a
completely Inteqrable system since the da. are congruent to zero (mod ui,,uu,
...,ω ) (see Note V-) Let x\x2t...,xn be a system of η first
П -ι 1«-И
Integrals Independent of this system, and let us denote by у ,...,t/
another system of r-n functions that together with the xb · form a family
12 η
of r Independent functions. The ω. are linear In dx ,dx t...tdx ; we
t
therefore obtain
(ω,)2 + (ω2)2 + ... + (ωη)2 ■ g^dJ
where the ?., may depend on the хг and the yl. Now let us set
"ij~ Xijkd*k+ "■ <*<#" -*j«J
12 и
where the unwritten terms are linear In dx\dx ,... tdx .
ь
By only writing the terms which contain the differentials dy , the
equations In (31) yield
K?|-
from which
3ω·
and finally.
3ω.
ω. —г· ■ λ.,,
t^T ал
W^**]
■
ωΛ-0
The form g.dcOx2 Is therefore Independent of the variables у . Q.E.O.

294
Rigid Displacement Group*
We have proved the existence of a group G of order r with η varl-
12 η
ables χ ,χ ,.,.,χ and the existence of an л-dlmenslonal Rleaannlan space
admitting G as a group of displacements; moreover, the group of rigid
rotations about a point Is the given group g.
288. There exists an Infinity of Rleaannlan spaces admitting the group
of displacements G with the stability group Imposed In advance. In fact
the group g which operates on the forms ω,,ω^.-.-.ω^ as an orthooonal
group could leave Invariant a certain number ν of linearly Independent
combinations of these forms that can be taken to be ω .,,...,ω -*
It Is also possible for It to leave Invariant several quadratic forms In
Μ,,ω^.-.-,ω . Let Q, .Q-,.. -1<) be these forms that are taken to be
linearly Independent. This being the case. It Is evident that the group G
will be a group of displacements for the Rleaannlan spaces with the fundamental
form
ο,ΙΙ, ♦ ^ ♦ ... ♦ аД ♦ «VjVyhV^ d.J " 1 Л »-v)
with constant coefficients α,,α,,.,.,α ,α.., subjected to the sole condition
that the fundamental form Is positive definite. Moreover, for all of these
spaces the stability group of a point will be defined analytically by the same
group g of orthogonal transformations.
Let us content ourselves by considering the case **4 and r-5 as 1n
the following example. Let
<L·, ■ -[им], day " [<*ιΰ] · Ли " ~[ω*ω] , Ли ■ [шли], Л) ■ О,
which correspond to
ω12 " ω34 " " · ω13 " ω14 " ω23 " "24 " ° '
The Infinitesimal transformations undergone by four forms ω. by an Inflnl-
t
teslmal rotation of g are
δω, x uu > δω- ■ -ω, , όω- ■ ω. , όω. ■ -uu
None of the linear forms In ω,, uu, ω , and ω. Is Invariant under g\ the
Invariant quadratic forms are
2 2 2 2
uj. + uu , ω- + ω. , ωιω3 + ω2ω4 ' ω1ω4 " ω2ω3
*The vectors e ,i,...,e and their linear combinations are at each point
я-ν ι я
the Invariant vectors under the stability group at this point; the number ν
Is thus the same as In no. 281.

Rigid Displacement Groups
295
and all of their linear combinations. Consequently there exists ал Infinity
of Rlemannlan spaces admitting the group G as the group of displacements,
and their fundamental forms are
2 2 2 2
Α(ω. + nu) + B(u£ + coj) + 2C(ui,uu + ay)*) + 20(ω,ω4 - оуч)
with the constant coefficients А, В, С, О satisfying the conditions
A > 0 » AB - C2 - 02 > 0 .
By a convenient choice of variables, the equations of the group are
(χ1)' β x1 Cos a - x2 Sin с + a}
(x )' -x Sin с + χ2 Cos с * a2
(x3)' ■ x3 Cos с - χ4 Sin с + с?
(χ4)' - χ3 Sin с + χ Cos с + α4
and In the fundamental forms Indicated, It Is possible to replace
1 2 3 4 . ^1^2,3,4
ω , ω , ω , ω by or , or » or » or
All the spaces thus obtained are locally Euclidean.
289. In fact we have. In a certain sense, solved a problem concerning
Rlemannlan spaces admitting a transitive group. The solution of this problem
has been reduced to several successive problems:
1. To determine all the orthogonal groups g uith η variables. This
problem was the subject of an Important paper by S. Medici,* who had solved It
completely up to η ■ 6. It Is now possible to have the full solution on
account of the work of E. Cartan on the linear representations of simple groups.
2. Given the orthogonal group gt to determine all the groups that adnit
g as tha stability group of a point. The above considerations give at least
the structure of the groups. If not the groups themselves; from this structure
the equations of the groups could be deduced by Integrating the ordinary
differential equations.
3. Given a group G, to determine all the Riemannian epaoes admitting G
as a group of displacements. This problem Is also completely solved If we
restrict our attention to the structure equations of the space. These
Immediately arise out of the structure equations of the group which Is essentially
the same. If the group Is In fact known, the Rlemannlan spaces are In turn
derived without Integration.
Mwi. Souota norm, sup Pisa, t. 10, 1908 (160 pages).

296
Rigid Displacement Groups
290. Finally we add without proof, that the Riemannian spaces admitting
a transitive group of displacements are analytic spaaee. Here we make a
convenient choice of co-ordinates and that to every given etruoture of the group,
and to every poeeible fundamental form constructed by the symbols ta,, there
oorreepondB an analytic normal, eimpVy-connected Riemannian space, and that
all the spaces admit the same expression for the fundamental form being locally
Isometric to each of them. The different analytic, normal and simply connected
spaces that correspond to the same group G, but not necessarily to the same
expression for the fundamental form, are homeomorphlc to each other.
IX. THE 3-0IMENSI0NAL SPACES ADMITTING A MULTIPLY TRANSITIVE GROUP OF
DISPLACEMENTS
291. Ue Intend clarifying the generalities of the preceding section by
studying three-dimensional Riemannian spaces admitting a multiply transitive
group of displacements.
The first problem to be solved Is to find orthogonal connected groups with
three variables. The case whereby we have the three parameter group of all the
rotations of ordinary space gives the spaces with constant curvature. We note
that In this case, the group of displacements of such a space Is the same for
the spaces whose constant curvature Is different, but with the same sign. This
2
Is because, simultaneously with a de , the group leaves Invariant all of
those that are obtained on multiplying It by a positive factor. There are1 thus
three possible groups, above all, distinct and which correspond to the three
classes of spaces with constant curvature positive, zero and negative.
292. Every subgroup of the group of rotations of ordinary space Is of a
one parameter type, and It Is the group of rotations about a fixed axis. There
then corresponds to this subgroup the class of three-dimensional spaces
admitting a four-parameter transitive group of displacements. These are the spaces
that we are going to determine.*
The rectangular frames adapted to the group will naturally be those that,
at a point К of the space, have their basis vector e. taken on the axis
of rigid rotations about this point. The relations In (27) of no. 283 are
then
«13 - «23 - ° (32)
*Th1s problem had been studied by G. Rlccl. Comptee vendue, t. 127, 1898,
pp. 344-346; L. 81anch1, Memor. Soc. ital. Soiense, 3e serle, t. 11, 1898,
pp. 267-352; G. Rlccl, Memor. Soc, ital, Scienza, 3e serle, t. 12, 1902,
pp. 69-92; С Rimini, Ann. Scuola norm, sup Pisa, t. 9, 1902 (57 pages).

Rigid Displacement Groups 297
and the equations In (28) become
(U14 ■ cui + bu>2 + си?
(33)
ω23 * αω1 + ^'ω2 + σ'ω3
If the method outlined In no. 258 Is applied, It may be seen that by an
Infinitesimal rotation of the group gt the components ω,, ω.» ьи, ω,3, ч^з
undergo the variations (with the factor ξ,- omitted)
He must
бок ■
^3
ω2
шш23
then have
ω23"
from which
л*»- "
δω- ■
^3"
Ьш, ,
-uij ,
■"13
■ω13
δω3 -
■
ж a'uu
0
- Ь'Ш] ,
с - σ' - 0 . α* - -δ , δ*
and consequently
■ Oil -j a OUi + Ьш«
(34)
ω23 * "^1 + αω2
The fact that the coefficients с and <?' are гего expresses a general
result that was proved geometrically In no. 284, that Is, the trajectories
of the vectors «3> Invariant under gy are qeodeslcs.
293. Now following the conditions In no. 286, let us form the structure
equations by putting ω,- " ω *
dum ■ -[ω«ω] + fl[u>,uu] + btuuuu]
du»2 ■ [ω^ω] - Ь[ш,ии] + α[ω-ω3] (35)
Л- x 2Ь[ш,Шо]
The exterior differentiation yields
[ω^Λΰ] - 0 , [ui,dd ■ 0 , abCuuuuuu] ж 0 ,
from which Is deduced

293
Rigid Displaommnt Οτνηφβ
l£o x ^[uuuu] · (36)
The Introduction of a new constant and exterior differentiation of (36) yields
ac[u>iuyiu] E 0
Finally, there remains three constants a, b, and c, such that the two
products ah and ac are zero. Two cases are then to be distinguished
depending on a being zero or non-zero.
294. The First Case, α j* 0, b - с - 0. We then have for the structure
equations
duii ж -[ω«ω] + a[ui,uu]
(37)
άω
3 w
as a о
with
ω13 = *>1 · ω23 ' *»Ζ *
Without changing the group, we can make a equal to one, by substituting
αω3 for u)3- The unique linear combination of «,, «2· «3»
Invariant under the subgroup σ Is ω-, and the unique quadratic form In ω,
7 7 I
and uu under g Is ш-, + uu.
The most general fundamental form of the Rlemannlan spaces that admit G
as a group of displacements Is therefore
de2 - A(u>* + u|) + Bug (A > 0, 8 > 0) .
In order to obtain an actual representation of the group G, we note that
on taking a" 1, the equations In (37) could be written as
а[ш, + itiiy) ■ [(ω, + iuu)(uu + tu>)]
<ί(ω- + ΐω) ж 0

Rigid Displacement Groups
299
We have one possible solution of the problen by putting
..- du + idv . f dx + idu
<*3 + ·« " ΊΓΤ77- · "ι + ίω2 " VTT^ !
fro· this we deduce
2 . 2 dr2 +
ω, + ни ■ —я—
u + ν
. »du + ve&
ω3 I .1
и + ν
ш'~т~т-
И + V
Finally, by putting
u ■ я cos θ , ν - я sin θ
we find that
ds - „
ω3 * Τ · ω - dB .
The most general Rleaannlan space admitting the group G as a displacement
group Is thus defined by the fundamental form
Je2 - *(** + Φ * ■*' (А>0. 8>0) . (38)
As for the group Itself, It Is obtained In a straightforward enouqh way, under
the guise
rx - k[x cos с - у sin с) + χ
у - к{х sin σ + t/ cos σ) + уо (39)
1 ь
with the four arbitrary constants χ , у , к > 0 and σ.
8y Interpreting χ, t/, я as the rectangular co-ordinates of a point.
It Is the group of similitudes that leaves Invariant the plane jb-0. The
fundamental form In (38) has constant negative curvature equal to - ί . 8y
replacing ж and у by /τ χ and /j у respectively. It Is the
anallagnatlc form (30) of no. 150, where the xy plane has been taken as
absolute. The space Is then the non-Euclidean space of Lobatchewsky, and Its
group of displacements Is that which fixes the sheaf of horospheres tangent
to each other at a point of the absolute (here, the point at infinity), or

300
Rigid Displacement Groups
that which leaves Invariant the non-Euclidean sheaf of straight lines. Here
the lines are parallel to each other In Lobatchewsky's sense, and they are the
orthogonal trajectories of these horospheres.
The property of the space having constant negative curvature Is recovered
without needing to find an effective representation of the group. Effectively.
If the forms
Ω..
are calculated
il. .
^ij+ Κ*ω./*]
we Immediately obtain
Ω12 = α ^ω1ω2^
Jl13 = а [Ш]ш3]
"23
α [ω^]
295. The Second Case. In the second case which remains to be considered,
it is necessary to take a = 0. 8y then taking ω - ύω. as the fora ω. we
have, on account of the equations In (35) and (36).
duj.
-[ujgu]
dbiy =
^3 "
(ill я
[ω,ω ]
ЗДЕииии]
-u[w^2]
(40)
where the coefficient с Is not the same as that In (36).
2 2 2
Again It may be seen that the form u£ + ш- Is the de of a surface,
by using the theorem In no. 287, and this surface has constant curvature с
We could moreover reduce b and с to fixed numerical values.
In the first case It Is possible to reduce с to one of the values 1,0
or -1. As for the coefficient bt If It Is non-zero. It could again be
reduced to 1; this gives essentially six distinct cases (six distinct groups).
The Rlemannlan curvature of the space Is easily determined. Me have
ш_- ■ ω + £ω- ,
ш13 ■ Ы2 ·
ω23 ' '^1
from which
Ω12 " **λ2 * ^ωΐ3"23^ " ^ " σ^ωΐω2^
fl13 ■ Α»13 " [«1г"23] С'Ь [ω1ω3] ·
"23 " ^23 + ^ω12ω13^ " ~Ъ ty*^

Rigid Displacement Groups
301
The equation of the Rlemann quadrlc (see no. 170) Is then
b2(X2 + Y2) + (« - 3b2)Z2 * 1
2 2 2
If the Riemannlan curvature с of the ώ , represented by ωϊ + uu.
Is less than 3b , then this quadrlc Is a hyperbola of revolution with a sheet
whose asymptotic cone has an angle at Its vertex greater than 120° If a Is
neqatlve, equal to 120° If a fs zero, and less than 120° If о Is positive.
2 2
If о - 3b ■ we have a cylinder of revolution; finally, if σ > ЗЬ » we have
an ellipsoid of revolution, firstly elongated then flattened; the Intermediate
state where the quadrlc Is a sphere corresponds to a1 4b (a space with
positive constant curvature).
These results are exact If b Is non-zero. For b * 0, the Riemannlan
curvature is zero for all the tangent plane elements with vector e3; for
all the other planar elements, it takes the sign of c.
296. As for the realisation of the fundamental form, this Is quite
straightforward.
We could take
ω2+ω2 = Α 4 %*¥ л л (А > 0. K-lf0or-l)
1 2 [i +jbr + yz)T
ω . d, + 8 «^ - ψ g (K - 1, 0 or -1) .
1 + 4 (x + У )
The constants b and <? of the relations In (40) are connected to the
constants A, 8 and К by the relations
and we have the fundamental forn
2 A(dr2 + ^2) + {[1 + | (χ2+ *2)]ώ + Ъ{хау - yds)}2
[1 + f (x2 + у2)]2
(41)
(K = 0, 1 or -1) ;
the equation of the Rlemann quadrlc becomes
82(X2 + Y2) + (KA - 3B2)Z2 = A2 . (42)
The space has constant curvature In both cases. I.e.
1 К * 0, 8 - 0 (Euclidean space) ;
2 К - 1, A - 482 (spherical space)

302
Rigid iHaplaoemant Groups
297. It Is Interesting to look at the form of the normal slnply
connected space In the different cases where It night arise.
Let us assume first of all that B-0; this signifies that the geodesies,
tangent at each of their points to the axis of rigid rotations about this point,
form a congruence of normals to the surface я - constant. In this case the
space Is the topological product of the straight line and a two-dimensional
Rlemannlan space with constant curvature. In other words, these spaces are
homcomorphlc to the topological product of the Euclidean plane and the straight
line. I.e. Euclidean space, If К ■ -1 or 0. They are honeomorphlc to the
topological product of the sphere and the line If К - 1.
If В Is non-zero and К ■ 0 or -1, then the result Is the same. This
Is no longer the case If К * 1; we shall see that In this situation, the
space Is homcomorphlc to a spherical space. In effect, since the coefficient
b of the equations In (40) Is non-zero and the form ω + -Sr ω- Is an exact
differential dp, then we might profit by taking froo the family of frames
adapted to the group, a family characterised by a single frame attached to a
point In the space. It suffices to constrain the co-ordinates of a point and
the parameter v, on which depend the frames attached to the point, to satisfy
the relation p«0. This Is possible as dp Is not a linear combination of
ω., uu and ω.; the function ρ depends effectively on the parameter v.
With this established, the equations in (40) become
By then putting
2 - . 2 - m 4b-
ш1 ' Τ ω1 ' "2 д "2 · ω3 Τ ™3 '
or which amounts to the same thing (not forgetting that К ■ 1),
ω. ■ 2Д uu , uu ■ 2Д uu , uu ■ 4B ω-
we obtain the relations
du-, ■ 2[й)2шз] ι ^Ц " 2[ΰΰω, ] , Д>- - 2[u7,uu]
There are the relations In (22) of no. 2B1 that define the spherical space
with curvature 1.
The Riemannian арааеа in (41) for which K- 1, В/О, are homeomorphia
to a epherioal apaoe. By referring back to no. 281, we can see that It Is
possible to replace the form In (41) of the ds of the spaces In question by
the form

Rigid Displacement Groups
303
ds ж 4A{drn + dx-, + dx0 + dx->)
0 ' l 3 (43)
+ (16B2 - 4A)(xodr3 - Xjiie + χ,ώ:2 - x^,)2 ,
where the χ. are related by equation
χ + χ. + х- + x, x 1
О I £ J
Theorem. The three-dimensional normal, simply-connected Riemannian spaces that
adnit a four-parameter group of displacements are homeomorphic to either a
Euclidean space, the topological product of a sphere and a line, or to a
spherical space. All of these spaces, whose Riemannian curvature is positive
everywhere, are homeomorphic to a spherical space, but the converse is not true.
29B. All the spaces obtained the two situations which can arise admit
a four-paraneter transitive group of displacements, but this group Is not
always the maximum group of displacements of the space, since some of the spaces
have constant curvature. Ue have thus solved, In short, the problem of
determining the four-parameter displacement groups In a space of constant curvature.
The first case (see no. 294) gives us the spaces with constant negative
curvature, where the four-parameter group Is that which leaves Invariant a bundle
of straight lines, parallel In the sense of Lobatchewsky. The second case
gives us the Euclidean space and spaces with constant positive curvature. For
Euclidean space, the four-parameter group Is the one leaving Invariant the
sheaf of parallel straight lines. Ue leave to the reader to prove that In an
elliptic space, It Is the same group that leaves Invariant a sheaf of Clifford
parallels (see no. 281), as the one resulting from the form (In 43) of the
2
ds corresponding to spaces homeomorphic to spherical space. It is
therefore possible to give a geometric definition common to the four-parametsr
displacement groups of spaces with constant, positive, negative or zero curvature,
with the Imposed condition of "parallel lines," be It In the sense of
Lobatchewsky, the usual sense of the word or In the Clifford sense, depending on whether
the curvature Is negative, zero or positive.
X. GENERAL INTRANSITIVE DISPLACEMENT GROUPS
299. In this section, we Intend showing that the determination of
transformation groups susceptible to an Interpretation as Intransitive displacement
groups In a Rlemannlan space amounts to the determination of the transitive
groups with which we have been concerned In the preceding sections. To this
extent. It suffices to show that If In an N-dlmenslonal Rlemannlan sapce, the
trajectories of an Intransitive displacement group are n-dlmenslonal, then It

304
Rigid ^Displacement Groups
Is always possible to choose a system of co-ordinates such that the first η
co-ordinates are transformed under the group, then the remaining h a N-n
are Invariant under this group.
12 h
Let us denote by и\u ,...,u the parameters which specify the
trajectories and take a particular trajectory along with a particular point 0 of
this trajectory. Let us consider the geodesic manifold at 0 formed by the
normal geodesies at Σ , stemming from 0; this Is a fc-dlmenslonal manifold
V.. We take h tangent vectors to this manifold at 0, such that the
coordinates of a point Infinitesimally near to 0 In V,, Is considered with
respect to these h vectors taken to be basis vectors of a Cartesian frame
12 h.
tangent to V.; let them by du\du t...tdu . As all the displacements of
1 2 Λ
the space fix the point 0, the components du\du 9...9du are fixed.
Consequently the h basis vectors In question are all fixed and because they are
linearly Independent, we arrive at the following theorem:
Theorem. Every vector stemming from 0 and normal to the trajectory Σ Λ ie
invariant under every dieplacement of the group G that fixes the point 0.
300. An arbitrary displacement taking the point 0 on Σ to a point
on Σ will In turn take the manifold V. through 0 to another manifold
ο η
A which will be the locus of normal geodesies at Σ stemming from A. All
of these manifolds fill out the entire space, at least In a sufficiently small
neighbourhood of Σ and admit a framework for the different points of the
space. For this It suffices to Introduce In I, a system of co-ordinates
1 2 π °
χ\x ,...,i and by convention, assign to the points of the manifold V,
12 η
stemming from the point A of Σ , firstly the co-ordinates χ ,χ ,...,x
о 1 2 h
of the point A and then the co-ordinates u ,uι,.../ of the trajectory
on which the point In question Is situated.
This established, the co-ordinates x' ,x ,...,xn of an arbitrary point
of the space are transformed under the group G In so far that It operates
on the points of Σ .
r о
Theorem. Given an {n + h) dimensional Riemannian epaoe transformed by a group
of displacements whose trajectories are n-dimenaions, it ie abxzye possible
12 и h
to choose a system of oo-ordinatee χ tx ,,.,,ι ,ил...ли such that the first
η co-ordinates are traneformed transitively amongst themelvee, the remaining
u co-ordinatee remaining invariant. *
d
*Cf. an Important paper by G. Fublnl [Artnali di Mat, 3 series, t. B, 1903,
pp. 39-81), where the framework of the points In space Is similar, but
nevertheless distinct. In the theory of the structure of groups, the property In
question of the displacement groups of Rlemannlan spaces Is expressed by saying
that these groups do not have ββββηΗαΐ-inuariantere·9· the group with two
variables χ and у and two parameters a and b defined by the equations
xl я χ + ay + b , У* в У

Rigid Displacement Groups
305
301. The result of the above theorem Is that the determination of groups
susceptible to being regarded as displacement groups of a Rlemannlan space
reduces to that of being regarded as transitive displacement groups. To the
equations of such a group, assumed to be In n-varlables. It suffices to add
an arbitrary number of h new equations and at the seme time, stating that the
h new variables are Invariants under the group.
The first part of problem I Is thus solved If It Is solved for transitive
displacement groups. The determination of Rlemannlan spaces admitting a group
of displacements susceptible to analytic representation by a given group Is
almost completely accounted for. Let G be a group of transitive displacements
of an n-dlmenslonal space and g the corresponding stability group. The group
g which operates on the forms ω,,ω0,...,ω as an orthogonal group (see no.
I с η
288) leaves Invariant ν linearly Independent combinations with constant
coefficients In ω. which can be taken to be ω .,,...,ω together with I
% π-ν·" ι η
quadratic forms Q1.Q0,....Q„ In ω,,ω0,...,ω which are Independent with
I £ I It rt-v
constant coefficients. With this established, the most general expression of
the fundamental form invariant under G Is
ds2 ·= AjMQjM + ... + A£(u)q£((d) + Ψ(ωη-ν+1 ωη,ώι\ώι2 duh)
12 h
ψ being a quadratic form whose coefficients only depend upon u ,u ,...,u .
Naturally, we must assume that Α,,Αρ,.-.,Α. and the coefficients of the
quadratic form ψ are functions of u',u2(...,ufc such that the ds Is
positive definite.
302. The locus of points whose first η co-ordinates are fixed are the
geodesic manifolds V^(i) at the point where they Intersect the trajectory
Σ orthogonally. But, In general, they are not totally geodesic and no longer
Intersect orthogonally the other trajectories Σ.
If, on the other hand, we consider the manifold V ., arising out of the
geodesies stemming from the point 0 of Σ whose tangent vectors at 0 are
Invariant under the stability group of 0, that Is, the subgroup g of rigid
rotations about 0, then this manifold Is totally geodesic on account of the
reasoning In 284. This Is true for all points engendered by the geodesies
passing through this point, and which are invariant under the same group g
of rigid rotations about this point. It will then contain all the normal
geodesies at this point to the trajectory Σ which contains It; the manifold
V +. will thus Intersect orthogonally the trajectory Σ.
Theorem. If ив construct the manifold V +. 3 engendered by the invariant
geodesies under the group of rotatione about a point in question of the spaceл
then this manifold is totally geodasio and interseots orthogonallyл all the
other trajeotories of the displacement group.

306
Rigid Displacement Groups
The theorem In no. 284 is a particular case of this theorem.
303. In the case where the trajectories are Individually transformed
under a simply transitive group, we have v-n and the manifold V .. becomes
the entire space. The manifolds V, are not. 1n general, totally geodesic.
If on the contrary ν*0, that 1s, 1f the sub-group g of rigid
rotations about a point changes each tangent vector at this point to the
trajectory containing 1t, then V . 1s Identified with V., and we have:
Theorem. If the stability group of a particular point in space ohanges eaah
direotion tangent at this point to the trajectory Σ containing itM then the
manifolds V. arising from the normal geodesies at this point to a particular
trajeotory are totally geodesic and intereeot all the trajectories
orthogonally.
XI. DISPLACEMENT GROUPS WHOSE TRAJECTORIES ARE LINES OR SURFACES
304. If the trajectories of a displacement group are one dimensional, we
know (see no. 255) that the group 1s of the one parameter kind. Here n- 1
with a single form ω, and out of necessity, with an exact differential that
1 2
may be taken to be ώ . The most general form of the de admitting a one
parameter group of displacements 1s
ds ■ g*dx dor
where the coefficients д.. are Independent of χ .
305. If the group trajectories are two dimensional, the group 1s of the
2 or 3 parameter kind.
If G 1s a two-parameter group, 1t 1s simply transitive and could be
1 2
defined as the set of transformations leaving Invariant two form ω and ω
1 2
constructed by the two variables χ and χ and their differentials; these
satisfy the structural equations
л 1 лг 1 2Ί , 2 вГ 1 2Ί
dui α Α[ω ω J αω * Β [ω ω J
There are two possible cases:
1° A ■ В - 0. In this case ω and ω2 are exact differentials that
1 2
are assumed to be ώ and dx , the group (which 1s abeHan) 1s
(x1)* ■ x1 + a [**)' - x2 + b ;

Rigid Displacement Groups
307
The most general corresponding fundamental form is
ds2 - д..ах1сЬр ,
Where the coefficients д.. are functions of the h - n-2 co-
3 4 и ад
ordlnates χ tx ,.,.,χ . The trajectories are the surfaces with zero
R1emann1an curvature.
η 9 1 9
2 А г1 0. By taking Αω - Βω as a new form ω , the first form
being preserved, we return to the case A « 1, В ■ 0. We could take
1 dr1 2 dx2
ω ■ !=γ , ω « **%-
X Χ
with the group
(χ1)' -oar1 + b, (x2)' - ax2 .
The meet general corresponding fundamental form is a quadratic form
dx* ,2 3 4
in —χ , —ж- , dx , dx t...tdx with independent ooeffiaiente in
X X
1 2
χ , χ . The trajectories are surfaces with constant, negative
R1emann1an curvature.* The group 1s not abeHan.
306. Let us now take the case of two-dimensional trajectories with a
three-parameter group. The trajectories are then surfaces with constant R1e-
mannlan curvature. The group 1s the same as that for a surface with constant
curvature equal to 1, -1 or 0 (see no. 291). This being the case, the spaaes
2
in question will have their ds of the form
ds2 - A(u)dd2 + g^tu'dJ (i,j - 3,4 n), (44)
9
the da denoting the fundamental form of a surface with constant curvature
1, 0, or -1 and the g*, depending on и ,u ,...,u alone.
The manifolds V, are totally geodesic and Intersect the group
trajectories orthogonally as an Immediate consequence of the relation 1n (44).
307. Rectaгк I. The results obtained 1n this section supply us with all
the three-dimensional R1emann1an spaces admitting an Intransitive group of
displacements. The one having three parameters corresponds to the fundamental
form 1n (44) which could be written here as
du2 + A(u)dd2 . (44)
*Th1s curvature could vary from one trajectory to another.

308
Rigid Dieplaoemont Groups
Remark II. Every R1emann1an space that admits the group
xl ж ax + b , у1 ж ay (45)
as Its transitive group of displacements, admits a larger group since these
spaces have constant curvature. But If the group 1n (45) 1s completed by
adjoining h Invariant co-ordinates, then the R1emann1an spaces acfcrlt the
group so obtained as a group of dlsplaceaents; this, out of necessity, 1s
Intransitive and does not 1n general admit a larger group. Let us take for
example h* 1 and
de2 - A(*) ** ί* + 2B(.) ^-^- + 2C(a) *-& + D(.)d.2 .
у У У
-2 + , 2
Every displacement group 1n such a space leaves the form №— Invariant,
У
but 1f the functions В(л) and С(л) are also both non-zero, then at least
one of the forms — л =&- , 1s left Invariant and consequently so 1s the
other; the group cannot then be greater than a two-parameter type.
If, In contrast, a multiply transitive group G 1s taken such that every
R1emann1an space Ε admitting this group admrlts a larger displacement group
S' , and if the subgroup g of S of wstationm about a point in apaae do*a
not leave invariant each veator в terming from this point (v ■ D), then
every space E' admitting G as a group of Intransitive displacements will
also admit a larger such group G1. This relates to the fact that the
fundamental form of the space will be, up to a factor. Invariant under the sum of
the fundamental form of Ε and a form constructed by the parameters u1 of
the trajectories and their differentials.

Chapter XIII
ISOMETRIC RIEMANNIAN SPACES.
RIGID DISPLACEMENTS IN A GIVEN SPACE
I. ISDMETRIC RIEHANNIAN SPACES
3D8. Let us recall that two equ1-d1mens1onal R1emann1an spaces are said
to be ieone trie 1f there exists between these spaces a point
correspondence preserving the fundamental form, I.e., such that the distance
between two points 1nf1n1tes1mally near In one of the spaces corresponds to a
similar pair 1n the other space. Such a correspondence equally preserves the
arc length of curves, the R1emaun1an curvature at a given point In a given
planar direction, etc.
We have seen already In (no. 22D) that two spaces with the sane constant
R1emann1an curvature are isometric;* we have also found the conditions
for 1 sometry of two symmetric spaces. If It 1s considered at any
point whatsoever, In the first space, then the Rlemannlan form
■4wr°~"P
together with that of the second space (again taken at any point)
Imply that 1t 1s necessary and sufficient conditions for 1t to be possible to
change from the first of these scalar tensors to the second, a convenient
change of the Cartesian system of reference.
The Importance of the Rlemannlan curvature 1n the conditions of
1sometry in fact, arises out of the necessary and sufficient conditions
stated 1n Chapter X (no. 218), but the application of these conditions
required that it was possible to determine the geodesies of the two spaces.
The theorem 1n no. 219 yields another assertion, valid only for the analytic
Rlemannlan spaces; It only Involves one point of the space. But at this point,
one needs to know the components R. ... and all their виеа&зз-гие oovariant
derivatives.
309. Before considering 1n Its entirety the problem of determining
whether two equ1-d1mens1ona1 spaces are Isometric, we are golno to look
at a particular example but one that 1s sufficiently general.
*In this case, local mappings as for all of this chapter.
309

310
Isometric Riemcmnian Spaoes
Consider an η-dimensional R1emann1an space such that, at least 1n a
domain 0 of this space, the R1cc1 quadrlc with the equation
R.-xV ■ 1 , (1)
has Its axes 1n distinct directions and moreover the extent of these axes (or
rather the Inverse squares of these axes) are Independent functions of the
12 η
co-ordinates и\u u . Every other Isometric space would possess
at least one domain D with an Interior 1n which the same geometric
properties are necessarily realised. We are going to determine the conditions of
1 sometry of two such spaces.
Let us Introduce Into each of these two spaces a family of rectangular
frames having at each point basis vectors consisting of unit vectors taken
along the principal directions of the space. At each particular point A,
there will be 2n ways of choosing the frame, depending on the manner of
numbering the axes and the choice of the sense taken. Once a choice 1s made
at a particular point, 1t will determine by continuity, the choice mede at all
the other points of the domain (1f, as we shall assume, this domain 1s always
simply connected).
With respect to these frames, the components R.. of the R1cc1 tensor
are always zero for ifj·, the remaining и. I.e., Rii·*^ *nn are by
hypothesis distinct and Independent functions of the co-ordinates.
If the second space 1s Isometric onto the first there will exist at
each point of 0 a rectangular frame whose nature has just been indicated,
and such that the mapping which takes a point Η of the second space onto
the corresponding point Η of the first will take the frame (Rn) of the
second space onto the frame (R„) of the first. There will exist at each
point 1n the second space functions ΪΓ,-,,ΐΰο»-- »R which necessarily will
be distinct. Independent functions of the co-ordinates. Following this
mapping, assuming Its existence, we have the relations
Я.. в R.. (without summation) (2)
This already shows that the mapping, 1f it exists, 1s defined by the equations
1n (2) which permit taking the co-ordinates и of the second space as a
function of the co-ordinates иг of the first: 1t 1s only possible to have
isolated solutions of the problem.
But 1t 1s possible to determine the necessary and sufficient conditions
for the actual existence of the mapping 1n what follows.
310. The equations in (2) representing the mapping
tiff.. ■ <fR..
Ы. X.X-
(3)

Isometric Riemannian Spaces
311
Let us put
R..i. 1s the covariant derivative of the component R.. of the R1cc1 tensor
Μ.ΙΛ t-t-
1n the *-th principal direction.*
The napping preserves the forms ω1, as well as each of the differentials
dR..., and the resulting equations are
\i\k " Ri£l* (without summation) (4)
It is therefore necessary for the trapping, that the equations in (2) should
imply the equations in (4). These necessary conditions are also sufficient.
In fact, the equations 1n (2), by hypothesis, Imply those 1n (4) and
automatically, the equations 1n (3) Imply the relations
R££|fc (ω*-ω*) - 0 (£ - 1,2 η) .
—1 1 ., η
Now the determinant of these η linear equations 1n ω - ω ω - ω
1s non-zero, for otherwise the η functions R11(R00 R would not be
\\ LL ПП
Independent. It can then be seen that the relations 1n (2) Imply
ω * ω ,
hence the equality of the two fundamental forms.
311. The mapping of the preceding section Involves algebraic operations
that could be partly avoided by substituting Into the η functions Riι»R22*
■■■tR » the following scalar tensors which are rationally calculable with
respect to arbitrary system of Cartesian frames,
A, - R*A- ■ R.^'r/.A- - r/r/V A - R. 2R
1 г * 7 * J 3 г о к η -L-,
These tensors are none other than the sums of the same powers of the η
functions R.-.** The theorem may be stated 1n the following way:
Theorem. In order that tuo n-dimensional Riemannian spaces (for uhioh the η
scalar tensors A, ,A« A are independent functions of the co-ordinates)
are to be isometric, it is necessary and sufficient that the point
*e.g., DR,, - <fR,, + 2R, ли.. Now the only non-zero component 1s R,. and
the corresponding factor ω,. - ω,, 1s zero. In fact R,, 1s a scalar tensor
(Irrational).
**To account for this, 1t suffices to see what happens to the tensors when
rectangular frames are employed, having as their axes the principal directions
of the space.

312
Isometric Riemannian Spaces
correspondence that represents the equality of these tensors in the two given
spaces also represents, via a convenient correspondence between the principal
2
oriented directions of the two spaces, the equality of the η oovarbant
derivatives of the η tensors in the η positive principal directions of the
two spaces.
lie shall not dwell upon the algebraic problem that would Indeed remain
to be solved by the application of this theorem.
Above all let us bear 1n mind the remarkable conclusion that knotting
the Riaci tensor and its derived tensor* suffices for recognising the
isometry of the tuo spaces when the tensors (in 5) of these spaces are
independent functions of the co-ordinates. But for the most general case
the conclusion would not necessitate the consideration of the R1cd tensor
and Its derived tensors as being sufficient for recognising the 1 sometry
of the two Riemannian spaces. Otherwise, two equi-d1mens1ona1 spaces with
the same constant curvature of the second kind would be Isometric, one
onto the other, which 1s not the case.
II. AN ANALYTICAL PROBLEM
3lZ. In order to find all isometric mappings between Riemannian
spaces, we first need to solve the following analytical problem:
Problem: Given two systems of linearly independent differential η forms,
the firet constituted by the forme
ω {utdu)t ω [utdu) ω {u,du)
12 η
constructed by the η variables и ,u >...,u and their differentials, the
second by the forms
-1 -2 -и
ω (υ,<Μι ω (υ,<Μ,...,ώ (\>,<M
12η
constructed by the η variables υ ,υ , ...,\> and their differentials, can
we determine if it is possible to express the variables υ as functions of
the variables и in a way that realises, in turn, the equality of the forme
of the two systems, and determine these functions?
313. Let us take the exterior differentials dui1 and d? forms of the
given forms and express the first as exterior quadratic forms of the η forms
ω\ш ,.,.,ω" taken to be Hnearlv Independent by hypothesis, and the second as
—1-2 -n
exterior quadratic forms of the forms ω ,ω ,...,ω :
♦Actually,they only Involve a part of the derived tensor.

Ieometria Riemarmian Spaces
313
(6)
г 1 2 и
For the ν , every choice of the functions of и ,u ,...,u , that results 1n
the equations Uit - ω1 «ИИ result 1n the equations с£>г ■ <£о\ and
consequently 1n the equation
*wN т cw>> ■ (7)
A simple case which we have already met (see no. 262), 1s when the С.,г are
constants. For the problem to be possible, 1t 1s necessary for the γ., also
to be constants, and there should be equality between each of the constants
of the two series. There 1s then an Infinity of solutions dependent on η
arbitrary constants.
Besides this simple case, let us first consider the extreme case where η
of the functions Cfc,t(u) are Independent. It would necessitate that the
functions Y,,,r(v] with the same Indices are equally Independent and, more-
и2/и ι\
over, that the A ; equations 1n (7) are compatible. The solution of
the problem, 1f 1t exists, 1s therefore unique, or at least the problem only
admits Isolated solutions. But by the argument that we have already used 1n
no. 310, we car see, by differentiating the equations 1n (7) and by putting
dCfchl " cfch*|t u*
that the equations 1n (7) must consequently Imply the equations
^U(v)-cw,V(u) ■ (8)
Conversely, let us assume that the equations 1n (7) are themselves compatible
and Imply those 1n (8). Consequently, they Imply the relations
Сл^|г(и)(£>*-</) ■ 0 . (9)
Now the rank of the matrix with η columns whose elements of the 1-th
column are the quantities С.,г..(и) 1s equal to η by virtue of the property
г ' h
of the functions C., (u) being Independent functions of the variables u .
The relations 1n (9) thus Imply
ω' (v,<fv) = ω {u,du)

314
Isometric Riemannian Spaces
and every solution of the equations 1n (7) provides a solution of the given
problen. These solutions are isolated.
314. Let us now proceed to the general case, and assume that amongst the
functions Cfc,t(u)I there are only η, < η that are Independent.
Let us assume that amongst the functions C,.1., resulting from the
differentiation of the С.,1, п0 are both Independent of each other and of
the Су. . Similarly for the functions C., ι. resulting from the
differentiation of the С.,1!,, η- are Independent from those preceding, and so on.
He will arrive at a point where, after a certain number ρ of
differentiations, a set of functions will be obtained such that a new differentiation
does not yield a function Independent of those preceding.
Two cases are possible:
1 The flotations so obtained nmtber η independently. It 1S necessary
and sufficient, for the possibility of the problem, that the equations
ykh\iMmCkh\iM
(10)
L ** IV2'"Vl llll2" p+1
are compatible, and this will suffice, as the equations 1n (10) on dlfferentla-
-1 1-2 2
tlon yield η linearly Independent equations 1n ω - и , ω -ω and
thus gives ω^ν,^ν) - uι (u.du). As there are η Independent functions of
the ν 1n the right hand sides of the equations 1n (10), the problem only
admits Isolated solutions.
2 The functions so obtained are in number υ < η independently. It 1S
again necessary for the equations 1n (10) should be compatible. Let us assume
that this 1s 1n fact the case. The differentiation of the equations 1n (10),
except for those that are 1n the last Una, will yield ν linearly Independent
relations 1n ω - ω , ω - ω ,.,.,ώ" - ω". Without loss of generality, we
may assume that these equations are solvable with respect to the ν remaining
««—« -n-v+l η-υ+1 -η η
forms ω - ω ,... ,ω - ω .
For the compatibility of the equations 1n (10), 1t will thus suffice to
add the compatibility of the equations
5*(ν.Α0 ■ J-{u,du) (i ■ 1,2 fi-v) (11)
between themselves and with those In (10).

Isometric Riemannian Spaces
315
Now on account of (10), the n-\> equations 1n (11) form a completely
Integrable system, the exterior differentiation of (11) yields the relations
YWi ΜΙϊ»(ν,ίίν)ω (ν,ίίυ)]
i к h (12)
■ C^ («)[ω {и,аи)ш {u^du)] ,
which are a consequence of the equations 1n (11) 1f we take Into account those
1n (10) and the fact that the differences i""v+1 - ω""4*1,...,ω1 - ω" are
zero on account of (11).
Finally the compatibility of the equation in (10) thus gives the necessary
and sufficient conditions for the existence of the problem, and the general
solution of this problem depends on η- υ arbitrary conetants. It can be seen
that the derivatives of the functions C..1 of order greater than n are not
considered.
315. For the applications, we shall have to conlsder the case where the
i it
coefficients of the forms ω are the same functions of the ν as the coef-
ff dents with the same Indices of the forms ω are for the u .
In this case the equations 1n (10) are always compatible, since they admit
the solution \>г - u . The problem will not admit a solution 1nf1n1tes1ma11y
near to the one for which the Integer υ 1s less than n, and the general
solution will then depend on n-v arbitrary constants. This amounts to
saying that the forms ш1{илаи) are Invariant under an Infinity of
translations depending on n-v arbitrary constants; these transformations manifestly
form a group. This remark will play the fundamental role 1n the determination
of the largest group of displacements of a given R1emann1an space (Section IV).
III. THE GENERAL ISOMETRIC MAPPING PROBLEM FOR RIEMANNIAN SPACES
316. We are going to relate this problem to the analytical problem
outlined 1n the preceding section. To this extent, given two equ1-d1mens1ona1
R1emann1an spaces, we attach to each point of one of them all the rectangular
frames having this point as their origin. For the first space these frames
depend on ^jj—*· parameters, of which η are the co-ordinates u^ of the
point of origin of the frame and the other "4" ' are the parameters ξ ,£ ,
. ..,ζ"*""1^2 that fix the orientation of this frame. For the second space,
those frames will depend on the same co-ordinates \>г of the point of origin
к к
and on the parameters η similar to the ζ that fix the orientation of the
sphere.
The forms ω. and ω., which define the elementary displacement taking
one frame to another 1nf1n1tes1ma11y close to the first space are constructed
with the co-ordinates иг and the parameters ζ1 as well as their

316 Isoimti*ia Hiemannian Spaces
differentials. But the η forms ω,,ω-,.,.,ω do not contain the άζ\ as
these are zero since the origin of the frame 1s fixed, I.e., since the
differentials du1 are zero. In contrast, the forms ω., could contain the dlffer-
h VJ
entlals dC : they are likewise linearly Independent with respect to these
nfn-1)
'g ' differentials, for 1f the forms ω, and ω,, are zero then this
signifies that the frame stays fixed, and that all the differentials du1 and
di, are zero. Similar comments naturally apply to the corresponding forms
ω. and ω., of the second space.
317. With this established 1t 1s clear that 1f two spaces are Isometric,
then every Isometric maDplna of one upon the other will correspond to а
frame of the first determining the second, and the mapping will transfer the
forms ω. and ω., to ω. and ω., respectively. The problem 1s thus
reduced to the following:
к ί
Problem. To express the funations υ, and η as funations of the и and
^ in a Day that satisfies the relations
ΰκ(υΛη,ίίυ) ■ ui£(u,£,du)
ω..(υ,η,<ί\>,ίίη) = цк .(u.C.du.dO
(13)
Conversely, every solution of these equations will yield for the co-ordinates
1 2 η
υ ,υ ,...,υ of the points of the second space, functions with co-ordinates
12 η
и ,u u of points of the first space since on account of the equations
1n (13), the differentials dvv are only linearly dependent 1n the
differentials du1. The point correspondence so established between the two spaces
will be a mapping since 1t will realise the eauallty of the fundamental forms
(ωΊ)2 + (ω2)2 + ... (шп)2 and (ωΊ)2 + (u^)2 + ... + (ω^2 .
forms which, moreover, do not depend on the variables n and ζ1.
318. He can see that we have returned to the analytical problem of the
preceding section, the number N of the differential forms being here n +
"i"p!l * *("+Ί). in order to apply the method that we have described, 1t 1s
cessary at
These are known; we have
necessary at first to calculate the exterior differential forms ω. and ω.,.
*»ΐ" Kv]
(14)
^ij ' [ωΐ*%'] + \ *ijkh Cu,ku,h] '

Isometric Riemannian Space
317
He see here that the first functions that appear and which play the part of the
C., of no. 313, are the components of the Riemannian curvature tensor. We
note moreover that the components do not depend on the co-ordinates иг alone,
but on the parameters ζ of the frames with respect to which they are
calculated.
We will then have to calculate the differentials of these components
RiiWi exPressed linearly with respect to the forms ω^ and ω... He
evidently obtain
^kh ' W*· + W^r + "Wife, + Κν*%· + hjkh\l"i ;
(15)
the quantities Ъ,.„\, being the components of the derived tensor of the
R1emann1an curvature, we will then obtain
ijk 1I rjkh 1t гг vrkh \ I jr
+ ν*|ι%· + 4/Ηι%· (Ί5τ)
*\jkh\r"ir**ijkh\lJm'
ЛЦкН\*я + RitfJtt | £mwir + hrkhlbrPjr
+ Rtffcr|WV + *ijrh\toft<r
(152)
+ hjkh\rm*lr + RtfJtt|*W
+ Ri^fc*|WV
and so on.
There exists an Integer ρ for which the right hand sides of the
equations (15) will be linear combinations of the right hand sides of equations
(15), (15,),...,(15 ,). This will signify that no component of the curvature
tensor ρ times derived 1s an Independent function of the curvature tensor
and of Its (p-1) first derived tensors. This Integer ρ being known,the
condition for isometry of the two spaces 1s the compatibility of the
equations expressing the equality 1n turn of the components of the curvature
tensor and Its first ρ derived tensors for the two spaces.
Theorem. Given an n-dimeneional Riemannian spaaet there corresponds to it an
integer ρ with the following proeprty: In order that a second space of the
вате dimension can be mapped onto it, than it is necessary and sufficient that
there exists between the two spaces a correspondence (rectangular frame to
rectangular frame) which realises the equality in turn of the components of

318
Isometrics Riemannian Spaces
their Riemann-Christoffel tensors, as well as their derived tensors of the ρ
first orders.
319. The Integer ρ is equal to 1 1f the components of the Riemann-
Christoffel tensor are independent functions of the η*ί ' variables и1 and
к
ζ . This is the case, for example, in the spaces considered in section I (nos.
308-311). In fact, to say that the coefficients R-ii.Roo R of the e<iua"
tion reduced to the Ricci quadric are independent functions, is to say that
the п*пл ι components R..(if»0 of the Ricci tensor are not bound by any
relation. The general theorem in no. 318 tells us that the conditions for
1sometry of the two spaces of the above nature only involve the
components of the Riemann-Christoffel tensor and those of its derived tensor,
when the theorem in no. 311 only involves the Ricci tensor and its derived
tensor. In concluding, it is necessary that the conditions stated in the
general theorem of no. 318 should not all be independent; we shall see other
examples of this in no. 321.
320. In the case of two symmetric spaces with the same number of
dimensions, the theorem of no. 318 gives a necessary and sufficient condition for
a particularly simple capping, i.e. that there exists a correspondence between
the two spaces (rectangular frame to rectangular frame) realising the equality
in turn of the Riemann-Christoffel tensors of the two spaces. As we know from
before, it is possible in each of these spaces to introduce a rectangular frame
at each point such that the components of the Riemann-Christoffel tensor are
constants, the condition of isometry being quite simply that with the
choice of frames made in one of the spaces, it is possible to make this choice
in the other space, so as to realise in turn the equality of the components of
the two tensors.
321. In the case n-2, we recover the classical conditions for a
mapping between two surfaces.* By taking К to be the Riemannian curvature,
the equations (15), (15-,), and (152) become
*See 6. Darboux, Leoons sur la th&orie dee surfaces, t. Ill, Livre VII, Chap.
II: E. Cartan, La thiorie des groupes finis et continue et la Geometric dif-
ferentielU, Chap. XII, no. 195, pp. 227-230.

Iaometria Riematmian Spaces
319
dK
dKl
dh
dKn
dK12
dK22
"
" Viz
= -Κιω12
- 2Κ12ω12
■ <κ22-κιι,"ΐ2 +
- -2K12|Ul2
■S·*!+ "rt ·
Κ11ω1 + "lrt
κιΛ ♦ κ^
Κ111ω1 + Ήΐ2"2
hz\^ * ^гг^
"22l"l + 4v!*Z
(16)
The six quantities K, K,, K_, K,,, K,2, Kg2 are functions of the
coordinates u, ν of a point of the surface with parameter ζ, which defines
the orientation of each frame having this point as Its origin. Thus there
1 2
exists five functions of these six arguments that only depend on и and и .
There 1s firstly the R1enann1an curvature К Itself. It 1s not difficult'to
find the other four; they are
ΔΊΚ- (ΚΊ)2 + (Kg)2 ,
ΔΊ(Κ.ΔΊΚ) ■ 2(Κ2Κ1Ί + гк^К^ + φ^)
■ Κ1(Δ1Κ)1 + Κ2(ΔΊΚ)2 ,
Θ(Κ,ΔΊΚ) - 2[Κ2Κ12 + ^KgtKgj - Κ1Ί) - ΐφ(12]
- ΚΊ(ΔΊΚ)2 - Κ2(ΔΊΚ)Ί .
Δ2Κ- Κ1Ί +Κ22 .
Δ, and Δ2 are the symbols of the two Beltrami differential parameters,
ΔΊ(υ,ν) - U]V] + u2V2, and the function e(U.V) 1s defined by the relation
[iiUfV] - eOl.VJCu^]
322. 111th this established and putting aside those surfaces with constant
R1emann1an curvature which are Isometric 1f they have the same curvature,
then several cases are possible.
1. The functions К and ΔΚ, are independent. The functions of the frame
K, K,, Kg are then equally Independent. The condition of nap-applicability
of two surfaces Included 1n this first case 1s the existence of a rectangular
frame to rectangular frame type of correspondence realising· 1n turn, the
equality of the functions K, Κ,, Kg, K,,, K,2, K^. If only point functions
are to be Involved, then the condition 1n question will be the existence of a
point to point type of correspondence realising the equality 1n turn of the
functions Κ, Δ,, Κ,-ρ Δ,(Κ,ΔΚ), θ(Κ,Δ,Κ), Δ^Κ. But, in practice, these

320
Isometric Riemomnian Spaces
conditions are superfluous. For brevity, let us put Δ,Κ - P. By putting
d? * Ρ1ω1 + Ρ2ω2 (with dK ■ Κ-ω. + K^)
1t can be seen that
2 2 2 (Pi^-*!^)2 + (P^-K-ifP)2
da = ω. + ω« ■ 5
12 (P^ - Ρ2ΚΊ)Ζ
ΔΊΡίίΚ2 - 2ΔΊ(Ρ,Κ)ίίΡίίΚ + ΔΊΚίίΡ2
(Ρ^ - Ρ2ΚΊ)Ζ
The napping will be confirmed 1f there exists a point correspondence realising
the equality 1n turn of the functions
Κ. Ρ, (ΚΊΡ2 - ly^)2, ΔΊΡ. ΔΊ(Ρ.Κ) .
2
but we cannot take account of the function (ILP2 - K-P,) since we have
(K7P2 - K2P7)2 = Δ,ΚΔ,Ρ - [Δ,ίΡ,Κ)]2 ,
whence the thearen:
Theorem. Given too surfaces for whioh the functions К and Δ.Κ are
independent, the necessary and sufficient condition for which these too surfaces
are isometric is the existenae of a point correspondence between the tvo
surfaces, realising, in turn, the equality of the functions
Κ,Δ^,Δ^Κ,Δ^,Δ^Κ)
In fact, put'lnp 1t more precisely, the surfaces are пар-applicable 1f
Δ,(Κ,Δ,Κ) and Δ, (Δ,Κ) are the same functions of К and Δ,Κ for the two
surfaces. It nay be seen that the twice differentiated components of the
curvature only Involve two functions Instead of three, the function Δ~Κ not
having to be considered.
2. The first differential parameter of К is a function of K. By putting
Δ,Κ - f(K), the following relations may be easily seen
ΔΊ(Κ,ΔΊΚ) - f(K)f(K). Θ(Κ,ΔΊΚ) - 0
> surfaces Δ.Κ 1s the same function of I
same for Δ,(Κ,Δ,Κ) and for Θ(Κ,Δ,Κ). With this in mind, we can subdivide
If for two surfaces Δ.Κ 1s the same function of K, it will then be the

Isometric Riemannian Spaces
321
2° into two other cases depending on whether Δ?Κ 1s or Is not an Independent
function of K.
a) Δ~Κ is an independent function of K. The functions with frames Κ, K^,
К-, К,,, K,-, Kp? are Independently, three 1n number and as a result of the
general theorem of no. 318, the conditions of nap-applicability of two surfaces
of the class 1n question will Involve the third covarlant derivatives of K.
By putting Δ-Κ ■ Q, we may see by a calculation similar to that made 1n case
1° that
de
2 Δ^Κ* - 2^{K,Q)d№) + &^W2
(ΚΊ02 - K^)2
and similar reasoning leads to
Theorem. Given tuo surfaces of the аЪаав in question, the necessary and
sufficient condition for them to be isometric ie the existence of a point <?or-
respondence between the tuo surfaces implying equality in turn for the functions
Κ, ΔΊΚ. Δ2Κ, Δ^Κ.Δ^), ΔΊ(Δ2Κ).
b) ΔρΚ, αβ uell as Δ, Κ is a function of K. By the general theory, the
conditions of map-app11cab111ty of two surfaces from this class only Involves
the covarlant derivatives of К of the two first orders. Every mapping will
be given by a point correspondence realising, 1 η turn, the equality of the
functions Κ, Δ,Κ, Δ«Κ, which again amounts to saying that Δ,Κ and Δ«Κ must
be the same functions of К for the two surfaces. There 1s then an Infinity
of mappings subject to the supplementary condition that they realise the
equality of the forms* Κ.ω* - Κ^ω-ι for the two surfaces. In fact, this
supplementary condition 1s necessary, and 1f 1t 1s realised, since we have already
assumed the equality of the forms Κ,ω, + Κ_ω? - dK for the surfaces, then we
will have the equality of the forms
*The form
Κ,ω« - Ко03 j
is chosen because to the nearest sign, it is
independent of the choice of rectangular frames, it 1s in fact the magnitude of the
bivector determined by the gradient of the function К and the vector Ά
2
By cat Una E, F and G the components of the de (Gaussian notation) we
have
**"*'*&'7*3
эк
Эй
1 2
Ыи +Fdu
ЭК
Эй
tdu + Ыи2

322 Iaometrio Riemannian Spaces
{^ + fyds2 ■ ΔΊΚΛ2 ,
and consequently the equality of the fundamental forms.
We might now note that there exists a function p(K) rendering as an
exact differential the form K^ - K^. By expressing the exterior
differential of the form pfK^ - K^) as zero, we arrive at the condition
ρ'(Κ)ΔΊΚ + ρ(Κ)Δ2Κ - 0 with £- * - ^
Let us choose a solution determined by this equation; we can then state the
following:
Theorem. In order that tuo surfaces of the class in question are
iaometrio, it ia neoeaaary and sufficient that Δ, Κ and Δ-Κ are the вате
funationa of К for the two surfaces. The mappings are then given by the
point correspondences that realise the equality of the Riemannian curvature
ae Dell ae the exact differential form ρ(Κ^ω- - K^) for t** tM0 surfaces.
The mappinga thus depend on an arbitrary constant and they are obtained by
quadratures.
The surfaces of this class are, as we know, Isometric to a surface
of revolution; furthermore» by putting
ΚΊωΊ + K^2 _ Λ . - η(κ ν χ „ ,
■ or» P(K^2 - K^l ■ ay >
we have
/E^t /K^t
as2 ■ ax2 + F2(x)dy2 with F(x) - Ί
p/b^
IV. THE MAXIMUM DISPLACEMENT GROUP OF A GIVEN RIEMANNIAN SPACE
323. The problem of determining rigid displacements of a given space
a particular case of the problem solved 1n the preceding section. Only 1n the
equations (13) of no. 317 is 1t necessary to assume that the forms ω. and
ω-, are constructed along with their arguments 1n the same way as the ω.
V 1 2 и
and ω., are with theirs. The variables ν ,v ν and the variables
λ 7 v3 „
и ,u u 1n the same Riemannian space are the co-ordinates of two points
that correspond to each other under a rigid displacement 1n this space. The
results so obtained now permit us to state the following general theorem:
Theorem. Given a Riemannian apace, it ia possible by simple differentiation
to reduce the determination of rigid displacements of this space to determining
the rectangular frame-to-rectangular-frame type transformations which leave

Iaomtria Riemannian Spools 323
invariant a certain meaber of parameters иг > ξ? that characterise these
franesa and a certain nunber of Pfaffian expressions constructed by these
parameters and their differentiate. The structure constants of the maximum
continuous group of displacements of the space are thus known without
integration. The functions and the Pfaffian forms to be considered only involve the
components of the Riemann-Christoffel tensor and its first ρ derived tensore,
ρ being an integer £ —Ц—*■ .
к
It is also possible to eliminate the parometere ξ by algebraic
operations (to deter/nine the invariants with respect to the orthogonal group of η
variables) in a way that needs only having to consider the point
transformations leaving invariant the functions of the co-ordinates и alone, and
the Pfaffian forms constructed by these co-ordinates and their differentiate.
324. Let us reconsider equations (15), (15,), (15J of no. 318. They
tell us the order r of the maximum group G of rigid displacements of the
space» as well as the order ρ of the maximum ηroups g of rigid rotations
about a point.
In effect» let ρ be the Integer correspondlno to the given space. The
number of linearly Independent forms with respect to the ω. and the ω.,
which figure 1n the right-hand sides of the equations (15), (15,)»..., (15- 1)1
1s equal to "'"a * - r. It 1s actually equal to the number of Independent
functions of the и and the ζ^ taken from amongst the components of the
R1emann-Chr1stoffel tensor and Its (p-1) first derived tensors, and this nunber
1s precisely the number of Invariants of the group G, considered as operating
on the rectangular frctnes of the space. Now let us consider the same right
hand sides of the equations 1n (15), (15,),...,(15 ,), but only leaving out
the terms 1n ω... The number of Independent linear foms so obtained 1s
equal to "(""') . Pt f0r at a generic point of the space, 1t 1s equal to the
к
number of Independent functions of the ζ that remain Invariant under the
group of rotations at this point.
The number of Invariants of the group, functions of the point co-ordinates
alone, 1s equal to the difference
-^.„.[ui^U.p],
η + ρ - г
It 1s the difference between η and the number of dimensions r-p of the
trajectories of the group.
tie can now deduce the number ρ and the orthogonal π roup of r1q1d
rotations of a point by a collection that could be called the Isotropy groups of
orders 0,1,2,... of the space at this point.
The Isotropy group of order zero 1s the group of rotations which leave
Invariant the components of the R1emann-Chr1stoffe1. The components ω,, of
the Infinitesimal rotations of this group are constrained by equating to zero.

324
Isometrio Riemcomian Spaces
the right hand sides of the equations 1n (15). where the terms wj are
suppressed. The order ρ of this Isotropy group 1s equal to "K" J minus
the number of linearly Independent forms 1n ω., which are Introduced 1n these
equations. The Isotropy group of order 1 1s the group of rotations that
leave Invariant the components of the R1enann-Chr1stoffel tensor and Its
derived tensor: Its order p, 1s the difference between n\""W and the
number of linearly independent forms 1n ω., which are Introduced 1n the
rlqht hand sides of the equations (15) and (15,). Similarly, the
Infinitesimal rotations of the Isotropy groups of successive orders 2»3,...,p-l are
obtained, as well as their orders p2,p3,...,p ,, which do not Increase. He
evidently have ρ , « ρ , and 1s the order ρ of the group of rigid
rotations at the point 1n question. He might remark that 1t suffices to let ρ
equal ρ , 1n order to arrive at the group of definitive rotations, because
on going from equations (15 _,) to (15 ), the nunber of Invariant points cen be
augmented.
325. Finally we add that 1f 1t 1s possible to determine without
Integration the order of the maximum group of displacements of a space as well as Its
structure, then 1η order to obtain the group effectively^ 1t 1s necessary to
Integrate a completely Integrable Pfafflan system. This amounts to Integrating
some ordinary differential equations (Note V) by Integration that could be
subjected to considerable simplifications following the structure of the group.
326. In the case of a symmetric space, only the equations 1n (15) are
Involved; the right hand sides of these equations only contain terms 1n ω...
There 1s no point 1nvar1ance, the group 1s transitive and Its order 1s equal
to Π'Υ ι minus the number of linearly Independent forms 1n ω·, that
occur on the right hand sides of the equations. These linear forms, equated
to zero, define the rigid Infinitesimal rotations at a point.
V. KILLING'S EQUATIONS
327. Nearly all the geometries that are concerned with finding the rigid
displacements of a given Riemannian space have comenced by determining the
infinitesimal displacements, achieved by Integrating a particular system of
differential equations constituted by what are known as Killing's equations.
We are going to construct these equations in this section. Given a Riemannian
space relative to a system of rectangular frames, let us denote by ζ. the
components of the elementary displacement undergone by a point N due to a
rigid Infinitesimal displacement. We denote by δ the symbol of this
displacement; the component £. 1s none other than the form ω. where the d1f-
ferentlals du of the co-ordinates are replaced by their Infinitesimal
Increments 6uk with respect to the displacements in question. He shall put

Ieometric Fienannian Spaces
325
ζ£ - ш£(б) and ξ€. - ω^.(δ) . (Ί7)
By using the symbols δ and d of differentiation (of which the first Is а
symbol of Indeterminate differentiation), the structural equations
could be written as
4 " *ΐ " h"kl " *«"*'
where
(18)
we denote by ζ.„ the fc-th «variant or absolute derivative of the tensor
From the equations In (18) we can extract the relation
\ &(de2) ■ и1€01й£- (ζ£Α + ξίΛ)ω£ωΛ . (Ί9)
The Infinitesimal displacement In question leaves invariant the fundamental
form and from (19), we find that the quadratic form of the last term Is
Identically zero and consequently the coefficients ζ.,,. + £.fc are antl-synmetrlc
with respect to the two Indices t and it. Now ζ.. * -ζ. . and we thus
establish
Theorem. In order that the vector field ζ. define в a rigid infiniteevral
displacement, it is neoeeeary and sufficient that its first derived tensor
Zj/i i* a bivector.
The reasoning was based on the assumption that the space was relative to
a family of rectangular frames, but the result Is Independent of the choice of
local frames. Generally, we shall have
ξί/;--ξ,Λ · (20)
328. The equations In (20) Involve certain second order differential
equations. Let us consider the case where Cartesian frames have been
arbitrarily chosen and we perform an exterior differentiation on the equations
jfi j. Jt i ι-i i

326
Ieomtria Riemannian Spaces
tfe obtain (cf. no. 192)
Ί JkA r h 1л m A r h 1Ί я Ί ,Α Α, λΓ h 1Ί
7 ^ЧыЬ" ω J ϊ |«[» ω 3 * 7 <* |W - ϊ |„)[ω ω ]
from which
ζ |ΐλ " e*|fci " ^WU 0Γ ζί|£Λ " 4\ht " *Чш
By carrying out two success 1vt cyclic permutations on the Indices i, К I we
obtain the three equations
4\Vi ' 4\n% " **ΗΗΙ
h\u' h\uM ^hoiii
from which we can easily deduce» on account of (20),
These equations are known as Killing18 Equations.*
In the Euclidean case» relative to rectangular co-ordinates x,, the
equations (20) and (21) are written
ЭС. ЭС. 32ζ,
1ST +af-° > ЗГЕТ"0 *
j г η г
Integrated» they yield
4 " aih + ai l°ij ' -aji]
the components a. relate to an Infinitesimal translation and the Ыvector
a.. to an Infinitesimal rotation about the origin.
*W. Killing, Ueber die Grundlagen der Geoemtrie (Journal de Crelle, 5. 109»
1892» p. 167); G. Rlccl» Sui gruppi oontimd di movimenti in una varieta
quainqua a tre dimeneioni (Men. Soc. I tab Sc., t. 12» 1899» p. 77); see also
N. Levy, Sur la oinSmatioue dee figures eontenuee eta* 1*8 eurfaoee oourbee et
en general done lee varietSa planes ou oourbee (C.R., t. 86» 1878, pp. 812-
816).

NOTE I
ON THE AXION OF THE PLANE AND CAYLEYIAN GEOHETRY
In dtaHng with the axiom of the plane we have made the Implicit
assumption that the geodesic surfaces of a Rlemannlan space satisfy certain
analytical conditions which» In turn» deserve closer attention. We are going to
assume In what follows that the coefficients g,. of the fundamental form
admit continuous first order partial derivatives: this confirms the continuity
of the quantltltes r, .. We shall assume moreover that these quantities art
endowed with properties that are strictly sufficient:
1° That the differential equations of the geodesies
2 i . , к , h
dw . г г du du я n m
da
admits one and only one solution corresponding to the given Initial conditions
2° That In a sufficiently small dona 1η of the space, there passes one and
only one geodesic between two given points.
He naturally put aside the determination of the analytical conditions
which must be satisfied by the coefficients д., for this to hold true. It
evidently suffices that they admit continuous partial derivatives of the two
first orders. With the hypotheses 1° and 2° established, we propose to show
how the Axiom of the Plane Implies the possibility of a geodesic representation
on ordinary space. We shall assume n-3 and write u, v, w In place of
Ί 2 3
U , U * U *
I. PRELIMINARIES
1. By putting
x ' uoe * * " vo8 » * " Wo* - (2)
where u't u\ W denote the Initial values of the derivatives of the unknown
functions, the equations of the geodesies stemming from a point A (u . ν ,
w ) could be represented In the form
327

328
Note I
У - vq - ?(ед,а) (3)
The relations In (3) define In short the representation of the Rlenannlan
space on normal Euclidean space (Chapter X),
Hypothesis 2° states that the equations In (3) are solvable with respect
to х,уля (for ц-и, у - у , и - и sufficiently snail),
2. Let us assume that In the region of the Rlenannlan space In question,
the coefficients Γ, are In absolute value less than a fixed number N.
He shall consider a geodesic arc situated In this region. With S as the
curvilinear abscissa, we obtain
u x μ β
о о
+ i β2ΐί"(θβ), 0 < θ < Ί
Now we have
u"(Qe) - -гД (u,y,u)[u'(e*)]2 _ ... ,
where или,и denotes the co-ordinates of the point on the geodesic whose
curvilinear abscissa Is θβ. Then u\ u\ w' take all possible values
compatible with the condition that the vector (u\y%u') Is unitary and that
at different points of the region In question, the right hand side remains
less than a fixed quantity ΛΜ, where h only depends on the coefficients
g. .. We then have
Ί Λ tu 2
- и и τ '
О
ц - ц_ ■ и'в + -к θ-|ΛΜβ
у " Vo " V'o8 + Ί е2Мв *4'
with
1 2
о о 2 3
|βΊ|. |β2|. |θ3| < ι
From this It can easily be deduced that If a displacement Is made on a qeo-
deslc surface at A(u ,v,u) and If I,, l?1 λ. are the covarlant components
of the unit normal vector to the surface at A, we have an Inequality of the
form

Bote I
329
\ΐλ{™0) + *2<ι>-ϊ>0) + Α3(^-ω0)| < feM[(u-uo)2 + {v-vq)Z
+ (liMJ ) ]
* о'
(5)
where к denotes a fixed coefficient.
3. Let us now consider a surface S containing a given aeodeslc (γ)
and which is geodeaio at every paint of (γ). We are going to show that the
unit normal vector to S at a point of (γ) remains normal to S when it ie
transported by parallelism along (γ). Ulthout loss of generality, we can
assume that the geodesic (γ) Is defined by the equations и - ν - 0, this
Implies, by virtue of (1), the equations
'3 3 '3 3 U
for every point of (γ). The covariant components of the unit normal vector
to S at a point of (γ) are of the form (ι,,Ι-,Ο). Ue wish to show that
at tvtry point of (γ) we have»
dh 1 2
AT" Vl 3-Vl 3" °
dU
~ *Т Гц ч ~ *л! л
~3ΰ" *V2 3 ' *2*2 3
■ νΛ - 12Γ323 " ° *
(6)
The third relation Is quite easily seen to be true. In order to prove
the other two» let us take a particular point A of (γ) for which we might
0. Ue could make a change of variables such that the quantities
are all zero at A (see no. 84) and without this, the equations of (γ)
u ■ ν - 0. It suffices then to prove under these conditions that
the point A, we have on displacing along (γ)»
assume w
'Λ
cease to be
dl, dt0
These two equations reduce to a single one by virtue of the relation
П.2 * ·> Ί2. . + 22.2 Ί
g l^ + 2g l^lg + g 1Λ - 1 »
which on differentiating and remaining at the point A» yields
11
.12. ^d*"\ ^ , 12.
22. χ dl2
b V',£*2>ТЬ7+<''V*"*2>ТЬ7-°

330
Note I
If, for example, we assume that at A, the equation of the tangent plane
element at
S Is du
dbi
0. (I. - 0), we will simply heve to show that
or that the ratio — tends to zero when ω tends to zero.
ω ,
Let us consider a particular geodesic (γ ) steaming from A and traced
across the surface S, with
Ί
0 ,
.1
Ь/Ό,
Ί
о f 0
Let us take a positive number η as small as we wish. There will exist
a domain D about A In the Interior of which the continuous quantities г
will remain less than π In the absolute value, Then following (4), we will
have for the geodesic (γ ),
и - χ θ^ληβ
ν - be + £ в^гца
ω ■ oe + j θ-ftna
As ω Is a function of s admitting continuous derivatives of the first two
orders. It Is the same as considering s as a function of ω and we could
write these by Introducing a fixed coefficient h t as
J *
и ■ θ',Λ'ηω
-ω + θ,'Λ'ηω2
о с
(7)
with
|θ"ίΙ»|β·2|
< Ί
Let us asslqn a fixed value
point M' of the geodesic value
w to
о
(Yl)
w and consider the corresponding
as belonging to the surface S
regarded as a geodesic to the point N(0»0»v ) of (γ). On account of (5) we
will have on denoting by I, ju and 0, the соvariant components of the unit
normal vector to S at the point H»
|ΐΊκ + l2v| < кт\(и2 + ν2) ,
where

Bote I
331
1*2 \ % + "4(ΘΊ11 + θ212}1 < **Ϊ(Ι + *F*o)2
: 21
шо\ »
* e-2h-2n2
alttrnatlvtiy
|| J I < V, lift β'ι2| + kn [(l^'n-J2 ♦ eiW»2]
о
Finally by Introducing a fixed number Η Independent of η, ω , l,, and
о I
λ.» we can write
Ισω0Ι
This Inequality shows us that for η however small a given positive number· It
Is possible to take ω so small such that the ratio — remains less than
ο ω
a l2 °
H|r|n In absolute value. This Indicates that -^ tends to zero with ω
when the point Η of γ tends to A. This Is what we wanted to prove.
4. From the preceding theorem we can straight away extract the following
result: If two eitrfaoee S, and S- intersect along a geodeeio (γ) and if
they are both geodeeiae at all of the pointe of (γ), then they intereeot at
a oonetant angle*
In fact the two unit normal vectors at S, and S~ At a point of (γ)
remain normal If they art transported by parallelism along (γ); consequently
the angle of the normals to the two surfaces Is constant right along (γ).
II. THE THEOREM OF F. SCHUR
5. A result arising out of the theorems which are going to be proved Is
that the tanqent plane element to a totally geodesic surface defined as as
geodesic surface at each of Its points varies continuously.
If all the geodesic surfaces at a point A are totally geodesic. It Is
easy to see that there always exists a geodesic surface at A containing a
given geodesic (In not too large a region about A). Effectively» let (γ)
be a geodesic and Η one of Its points and let (γ') be a geodesic passing
through A and H. He shall consider the unit normal vector at Μ to (γ)
and (γ1) and transport It by parallelism from Μ to A along (γ'). There
exists a aeodeslc surface S at A and normal at that point to the Ыvector
so obtained. The normal at Η to this surface, which ie a geodeeio along
(γ1), will be on account of no. 3, normal at M, not just for (γ1) but
also for (γ). The geodesic (γ) tangent at Μ to S will then be
completely contained In the surface S which Is geodesic at N.

332
Note I
We are now In a position to prove Schur's theoren following which the
space satisfies the Axiom of the Plane If there exists two points A and В
such that every geodesic surface at one of these two points Is totally geodesic
(see no. 112). As In no. 113, we first of all attach to each point Η of the
space the six quantltltes x, y, a; x\ yl, a'. The first three are defined
to within an arbitrary factor as well as the last three. We have proved an
Important relation between these quantities In the text by our reliance on the
property of the four totally geodesic surfaces through AB having the same
cross ratio at A and at B. The proof of this property given In the text
does not apply here. But thanks to the theorem proved In no. 4 this property
Is evident» since the angles at which the four surfaces Intersect at A are
the same as those similarly described at B.
He could then apply the general result recalled from no. 114 where It
first appeared. We can attach to each point of the epaoe four homogeneous
co-ordinates Χ, Υ, Ζ, Τ such that every totally geodesic surface through A
ie defined by a linear equation in Χ, Υ, Ζ and through B, by euoh an
equation in Χ, Υ, Τ. Consequently every geodesic being the Intersection of two
of these surfaces Is defined by a system of linear equations In Χ, Υ, Ζ, Τ.
Putting It another way, a Riemannian space admits a geodesic representation
on an ordinary space where the geodesies are represented by straight lines.
6. What Is no longer evident Is that planes In an ordinary space are
Images of geodesic surfaces In Rlemannlan space because the non-homogeneous
χ γ 7
co-ordinates γ, γ, у of the ordinary space are functions of co-ordinates
u, ν, ω of the Rlemannlan space whose nature we may ignore. Thus It Is not
certain that geodesies tangent to a common plane element In the Rlemannlan
space have as Images straight lines situated In a common place In an ordinary
space.
This property Is nevertheless exact at the point A and at the point B,
on account of the equations In (8) In no. 113. To prove the general case, let
us take a point Ρ of the Rlemannlan soace and Its Image P1 In an ordinary
space. Every geodesic stemming from Ρ Is defined by the common ratios of the
three quantities {duy dvy dw)\ the stralqht line Image will be defined In the
same way by the common ratios of the three quantities d£, dr\t dz where, for
Χ Υ Ζ
example, ζ, η, ζ denote the ratios γ, γ, γ . He regard du, dv, da on one
hand and d£, dr\, dr. on the other as homogeneous co-ordinates of two points
M, M' of a plane Π. ' Let a and Ь be points corresponding to the geodesies
PA and PB respectively; let a1 and b' be points corresponding to straight
line Images P'A* and P'B" respectively.
Every plane element stemming from Ρ In Rlemannlan space will be
represented In the plane Π by a straight line d and every plane element stemming
from P* In ordinary space by a straight line d'. If the line d passes
through a, the plane element stemming from P, tangent to a totally geodesic

Note I
333
surface passInn through A, has as Its Image a plane element stemming from
P' forming part of a plane passing through A1. Consequently for every line
d passing through at there corresponds a line d' through a*. Similarly»
for every line d в terming from Ъя there corresponds a line d* 8 terming
from b*. Finally, on account of the above remark, the cross ratio of four
lines d stemming from a (or from b) Is equal to the cross ratio of the
corresponding four lines d* stemming from a' (or from bl).
7. With this established let δ be an arbitrary side of the plane Π
corresponding to a geodesic surface Σ at P. Let us find the locus of points
m* which correspond to the different points m of δ. Let m be the point
where δ cuts ah and let m., m. and m- be any three other points of δ.
The cross-ratio of the four lines (a, m m, m- m.) Is equal to the cross
ratio at A of four qeodeslc surfaces containing the geodesic AP. It Is
therefore equal to the cross ratio of these four surfaces at Ρ and
consequently equal to the cross ratio of the four curves of Intersection of these
surfaces with Σ. We can evidently produce the same result by taking the cross
ratio of the four lines (b, m^ m^ ^2 "*
J. The result of this Is that we
shall also have
{a* m* ml ml ml) ■ {b' m* ml ml ml) *,
consequently the points m.', ml, ml are In a straight line. To 6 there
corresponds a straight line δ'.
The correspondence (M,M') In the plane Π possesses the property that
for every straight line there corresponds another and the result Is that to
every plane element of Riemannian space there corresponds a plane element of
ordinary space. Consequently, the surfaces of the Riemannian space which have
as their images the planes of the ordinary space are totally geodesic. The
Axiom of the Plane Is thus proved for Riemannian space.
8. Furthermore» the point correspondence (M,M*) of the plane Π Is
projective. Putting It another way, we have on making a displacing along an
arbitrary oeodeslc stemming fron a point P,
du dv dw
adi + bdr\ + adx. * a'dt + b'dn* ο'άζ " a»dt + b"dr\ + a"dr.
The cosine of the angle of both directions will therefore have. In the ordinary
space where the representation was made, exactly the sane form as in a system

334
Ifote I
of Cartesian co-ordinates for which the Isotropic cone would then be a cone
determined to the second order.
We could add, moreover, that If a displacement Is made along the straight
line dr\ ■ <ίς в 0, the quantities τ-ρ, τ£, 77 tend towards determinable
limits. Putting It another way, the co-ordinates u, y, w with respect to
Χ Υ Ζ
γ-, γ-, γ- admit first order partial derivatives and conversely.
The transition from this to Cayley's geometry Is no more difficult than
the reasoning In the text (nos. 156-157). Every Riemcmnian epaoe satisfies the
Axiom of the Plane and ie thus locally Euclidean, spherical or hyperbolio.

NOTE II
ON LINEAR RIENANNIAN CURVATURE
We have proved In Chapter VII that the Rlemannlan curvature arose from
the development of the space along a closed contour. For a contour bounding
an Infinitesimal area about a given point, this curvature depends on the
orientation It Is proportional to this area. It could be said. In short, that Rle-
nannlan curvature has a superficial magnitude (that Is to say, attached to a
surface element of the space).
To this extent we have assumed that the coefficients д.. of the
fundamental form admit continuous partial derivatives of the two first orders.
We are going to see that thlnos change when this hypothesis Is renoved.
Let us consider the simplest case where. In a certain dooaln In a Rlemannlan
space, the д.. adalt continuous partial derivatives of the two first orders
with the exception of points of a surface Σ traversing the domain. Me shall
assume that the д.. have at e^ery point on the surface continuous first order
partial derivatives, but with discontinuity of the normal derivative.
In a precise way, the function д., admits at a point N of Σ a well
defined derivative In all directions. But the derivatives taken along two
opposite directions normal to Σ do not have the sane value whilst the
derivatives taken alono two opposite directions tangent to ς Is defined by the
equation u ■ 0. To the two sides of Σ we shall asslnn the names positive
side and negative side. The Indices i and j fixed, the covarlant vector
Эк*
7*
. oU
Is by hypothesis, normal to the surface. Its first two covarlant components
[к - 1 and 2) are thus zero. If we call h., the abrupt variation of the
normal derivative when It passes from the negative to the positive side, we
have
3ffv
Эц"
1 +
3ffv
ди*
1
h.t
J.
/7* **
(Ί)
If we call H^ the difference between the two values of «... on the
positive side and on the negative side, we obtain without difficulty, the
va1ues
V
335

336
Mate II
Чу ■ °
Hi3j ■ - нзу -·\-$γ
H33i c ° ■ нзгз " ^r^
η -1 *33
W.*.J ■ 1.2)
UJ ■ 1,2)
(i ■ 1.2)
(2)
With this established, let us consider a small arc of the curve HH' traced
across the surface (Σ). Let us start with a vector Хг stemlng from N
and transport It by parallelism along the arc of the curve MC of the
negative side of (Σ) and then along the arc of the curve M'M of the positive
side of (Σ). For the first transition, the vector takes components Уг given
by the relations
Y£ - X* - Х*(Г*) - du* .
к jf
For the second transition, the vector takes the components lb given by the
relations
ll ш уг + γ* (r* ) + dur
* к г'
- X£ + XfcHfctr du* .
(3)
Let us note that on account of (2), the H,. . where the third Index ν f 3
satisfies the relations H. . ■ -H., . The result Is that the parallel
transport in question has subjected the vector to an infinitesimal rotation with
oovariant components
Α.. - Η., du
ij ijr
We immediately find
'12°°
13= -~^!ί ('ίΧ+'ί^2)
.°23a- 2/3ϊ (л^^гг*2) ·
(4)
It can therefore be seen that for every arc of the elementary curve traced
across (Σ) there is a rotation about an axis tangent to (Σ) and which
translates what could be called the linear Riemannian curvature of the space
in the direction of this arc of the curve.

Sot* II 337
Ί 2
If we denote by ds the arc-lenqth of a curve, by α . α the direction
parameters of Its tangent, a straightforward calculation show that the Interior
product of the vector МЙ1 and the blvector which represents this rotation.Is
equal to
^ΛΊ1(ΛΊ)2 + 2h12du*<iu2 + h22)du2)2]
■ ^>η(αΊ)2 + 2д12Л2 + h22(a2)W .
The scalar quantity
К - ^„(α1)2 + 2Λ12αΊα2 + *22(а2)2] (5)
Ί 2
Is the Rlemannlan curvature of the space In the direction (a .a ). This
expression Involves the discontinuities of the normal derivatives of the three
coefficients y,,, £,-, 922 wnicn define the netrlc on the surface. If we
take an Infinitesimal lenqth £ on the normals to the surface at the different
points of the arc of the curve, In each sense, we obtain two new arcs of the
curve with lengths da+ and do_. He have
do\ + do1 - 2ffe2
К - 11m —- =,
ε-*0 2εώϊ
da+ + do - Us
"11m ш
The curvature К is than the am of the coefficients of dilatation of an arc
of the curve traced across the surface when it ie displaced normally to the
surface in each of the too senses.

NOTE III
ON NORMAL SPACES WITH NEGATIVE OR ZERO RIENANNIAN CURVATURE
On dialing with normal Rlonannlan co-ordinates, we could establish some
noteworthy properties of the normal Rlonannlan spaces with variable Rlonannlan
curvature, In the case where this curvature Is negative or швго at every point
and In every planar direction.
I. PRELIMINARIES, PROPERTIES OF THE de2 IN NORMAL CO-ORDINATES.
1. In no. 56 we gave the definition of a normal Rlonannlan space. He
will make here a supplementary hypothesis which might perhaps be unnecessary.
He shall assume that In each part of the space, represented analytically by
means of a system of co-ordinates и , the coefficients g^, of the
fundamental form admit continuous partial derivatives of the first three orders,*
the form naturally being positive definite.
The Г. . then admit continuous partial derivatives of the first two
t j
orders and the Rj-w continuous partial derivatives of the first order.
By virtue of the classical theorems on differential equations, the
quantities иг - (и ) taken to be functions of normal co-ordinates (see no. 213)
о
χ' ■ ο'β,... jX ■ а в
relative to the point (иг) admit continuous partial derivatives of the
first two orders. It Is much the same as for the components of the vector
obtained on transporting by parallelism a fixed vector along a geodesic
stemming from the point (иг) .
о
2. With this established. Instead of considering directly the de2 of
the space expressed In terms of the normal co-ordinates, we are going to define
It as the sum of the squares of the projections of the Infinitesimal vector
МГ on the axes of a rectangular frame (R) conveniently chosen with origin
M. For this we shall commence as In Chapter X (no. 216), with a fixed
rectangular frame (R ) havlnq the point 0 with co-ordinates (u ) as Its origin
and we shall transport It by parallelism along the geodesic which joins the
point 0 to the point M. Naturally enough, this will only be possible If
there exists a geodesic joining these two points.
In place of two, as we assumed In no. 52.
339

340
Rote III
We shall then provisionally put aside that part of the space which could
not be touched by the geodesies storming from 0. Conversely, If there exists
several geodesies joining 0 to M, we shall attach to Η as many
rectangular frames (R), each of which will correspond to a system of normal
co-ordinates for H.
We shall denote by
ауа2.....ап
the direction parameters relative to the frame (R ) of a tangent vector at
0 to a geodesic stemming from 0; we shall not provisionally restrict this
vector to any particular degree of magnitude. We shall then have for the
normal co-ordinates of a point N on this geodesic
1 9
χ ■ a*t , χ ■ аЛ9...%хп ■ a t , (Ί)
such that the length of the arc 0И will be
s » t Ja\ + a\ + ... + a\ . (2)
We Introduce In the summation the (n+1) superabundant co-ordinates
ал,...,а ,t & 0
ι η
3. Let us denote by ω,,ω-,.,.ω the orojectlons (on the axes of the
frame (R) with origin N) of the Infinitesimal vector whose components with
12 η
resoect to the natural frame attached to Μ are du \du ,.., tdu . The ω.
Ί η г
are linear forms In du ,...,du with coefficients that admit continuous
partial derivatives of the first two orders.
The rotation about Η which takes the frama (R) to be parallel to the
frama (R1) attached to the point H' Inflnlteslmally near Is defined by
a Mvector, the comoonents of which are ω.. » -ω., relative to the frame
•ι \Q Jt
(R). These are linear forms In du du" with coefficients that admit
continuous partial derivatives of the first order.
When the oaramaters a. are fixed and t Is varied, the Infinitesimal
vector MH1 admits the a. as direction oarameters with resoect to the frama
г
(R), and the forms ω. thus reduce to
ω. ■ a.dt
г г
Similarly, on transporting by parallelism along the geodesic OH, we shall
have
ω. . ■ 0

Note III
341
As In no. 216, we shall establish
f ω. β a{dt + ω/
Ι ω,, ■ ω, .
Ι 4 tj
where the new forms ω. and
zero for t - 0.
•
ω
(3)
., are linear In da,tdaot...,da and are
ν ι с η
4. Let us start with the fundamental differential equations In (7) of
no. 217.
IT ώΐ + Vfct
3ω.,
-rib - R a ω
3t ijre r β
(4)
In principle, we are assured of the existence of continuous first order partial
derivatives for the coefficients of the forms ω. and ω... exoressed In
terms of the variables t, а^,...яап* But the first equations In (л) Indicate
that the forms ω., expressed linearly In da,,...tda , admit continuous
partial derivatives of the second order with respect to the variable t. If we
differentiate these equations with respect to t, we obtain, on account of the
last equations In (4),
^T"RHr.efcV. (5)
The result Is
2
Э ω
ω. —к— ■ R. . л,а ω .ω.
г 3tZ fctrj fc г г j
If It Is multiplied by the square of the magnitude of the Μ vector determined
by the two vectors a^ and ω{, the right-hand side represents the Rlemannlan
curvature with Its sign changed In the direction of the planar element of this
Ыvector. The hypothesis made on the sign of the Rlemannlan curvature of the
space then yields the fundamental Inequality
Э ω ·
ΰ{ -^ * 0 . (6)
The Immediate result Is

342 Sot» III
(7)
•»[£) · Gtf * - · Ш] ■
5. Considered as a function of t (the quantities a. and da. being
regarded as paraweters), the sua Tju,, which Is zero for fO, as well as
Its first derivative, has Its second derivative constantly positive; It Is thus
essentially positive. For a systen of values (t ,a,,da,), the conclusion
would only break down If, on account of (7), for all values of t between
Ям*
zero and t , we have -?— ■ 0.
О ot
Now the Initial value of this derivative Is, on account of (4), equal to
da.; It would then necessitate that all the da, became zero.
-2-2
The quadratic form ω, + ... + ω of the differentiate άα^άαρ.,.,άα
ie therefore poeitive definite for all values of the quantities t > 0,
αΊ V
6. Ue can go a bit further. Let us consider the function
which Is zero at the origin and whose derivative with respect to t Is equal,
for t-0, to /da* + ... + da\ Ue can easily verify the formula
«2-г , / Эш. Ьш.\
ω. i« i (ш, Ζί - ш4 -J)
» /7*7—Γ^ г IT7 , г у г тг j тг;
Нг V ' "' А7Т 7^? г"? * * -2Ί3/2
/ ω, + ... + ω L(i)7 + ... + ω 1
1 π 1 ηΛ
Ue, therefore, consistently obtain
/ω? + ... + ω* t ι4ζ? + dal + ... + da2 .
ι η ι £ η
or alternatively
ω2 + ω2 + ... + ω2 2: t2(da2 + do2 + ... + αα2) (8)
9
7. Let us now return to the normal co-ordinates хг. As the ds of
the space only depends on these co-ordinates. I.e., on η combinations ал%
г 2
we could obtain It by setting t - 1 throughout and a, - χ . The tie
9 9 *
then reduces to ωί + ... + ώ and the relation (8) becomes
ι η
de2i (dr1)2 + (dr2)2 + ... (drn)2 .
(9)

Bote III
343
We now restrict our attention to draw from this the following two
conclusions
1 At every point Η of the space, touohad by the geodesies stemming from 0,
the functional determinant of the normal co-ordinates χ with re ар sat to
the variables u\ ie nonzero, since it ie the square root of the quotient
of the discriminant of the too linear elemente, one expressed In terms of the
variables хг, the other In terms of the variables иг.
2 In the representation of the Riemannian epaae on the normal Euclidean space,
the Riemannian dtetanoe betueen two pointe infiniteeimally apart is greater
than or equal to their Euclidean dtetanoe.
We had already proved (In no. 227) this last property for an Immediate
neighbourhood of the point 0.
He could also constrain the parameters a. to the relation
2 2 2
a: + at + ... + a - Ί
I С П
We could easily show that we obtain
de2* dt2 + t2(da? + ... + da2) , (10)
ι η
where t would now denote the length of the geodesic 0И.
II. THE SIMPLY CONNECTED COVERING SPACE
о
8. With the aid of the de normal to the given Riemannian space which
we shall denote by ε. we can define a simply connected normal space ε*
honemorphlc to Euclidean space. It will be formed by the points N of the
space ε that are susceptible to being touched by the geodesies steeming from
0, but by regarding the same point N of ε as constituting several distinct
points of ε* If N could be touched by several distinct geodesies stemming
from 0. In other words, every point of ε1 Is the set of points N of ε,
and of a system determined by the normal co-ordinates (x ,...,*n) of this
point.
It Is clear that at a point of ε1 there corresponds a unique point of
ε. but at a point of ε there could correspond several points of ε1. We
are going to prove that every point of ε hoe at least one correspondent in
e', that is to say could be touched by a geodesic from 0.
9. Effectively, let us Join the point 0 to the point N by an
arbitrary continuous line (C). We could, at least In the neighbourhood of 0·
develop the line (C) on the space ε1, that Is, to follow continuously the

344 note III
normal co-ordinates (x ,...,xn) of the points of (C). This development may
be continued without ever stopping. Let us assume In effect, the contrary,
and denote by A the first limit point of the set of points of С to which
It Is Impossible to have a corresponding point of ε1.
The point A could not Itself be touched In the development unless It
would aotolt fixed normal co-ordinates (x ,...,xn). But as the functional
determinant of the хг with respect to the цг Is nonzero In a neighbourhood
of A, the development can be continued to all points sufficiently near to A,
contrary to hypothesis.
Although the point A 1s untouched In the development, all points prior
to A are touched. The normal co-ordinates χ ,...,χη of these points at
least a*n1t a system of limit values (x ) (xn) . On account of (10),
we have 1n effect,
(хУм*2)2*...*^)2*!.2 ,
where L 1s the total length of the line C; this proves that the set of
points (x ,...,x") 1s bounded.
It 1s, however, no longer possible to hav~> a Hwi4 point (хг) , for by
о
displacement along С before the point A, w> have on account of (9),
de2 & (dr1)2 + (dr2)2 ♦ ... + (drn)2
such that the Euclidean distance between two points (хъ) and (хъ),
corresponding to two points of С that are wry near to A, remains constant no
matter how small 1t 1s.
The result of this 1s that the point A could be touched 1n the
development, Its normal co-ordinates being (x ) . We thus arrive at a contradiction.
The space ε ie thus covered cotrpletedly by the geodeeioe stemming from
one of ita points.*
10. The space ε' 1s the covering space of the space ε. With respect
to ε, 1t plays the same role as Euclidean space with respect to the normal,
locally Euclidean spaces. It 1s simply connected and homeomorphio to a Suclid-
eab space.
In the particular case where ε 1s Itself simply connected (that 1s,where
e^try closed one-d1mensional contour of 1t 1s contractlble to a point by a
continuous deformation) 1t can be proved by a similar argument to that made 1n
Chapter III (no. 61) that to one point of ε there corresponds only one point
1n ε'. There- 1s therefore a bljectlve correspondence between the two spaces
See 1n Note IV, another proof of this theorem applicable to all normal R1e-
mannlan spaces.

Bote III
345
and consequently between the given space and Euclidean space with co-ordinates
(χ ,...,* ).
Consequently every normal, simply oonneoted Riemarmian враое with
negative or лето Riemannian curvature is homeomorphio to Euclidean space.
This statement 1s not a simple tautology. We could, for example, Imagine
a priori a space homeomorpMc to the volume between two concentric spheres (the
surfaces of these two spheres being at Infinity for the metric of the space).
One such space could not have a curvature negative or zero everywhere, for It
would be simply connected, but 1t would not be homeomorpMc to a Euclidean
space, since 1t would contain closed surfaces unsusceptible to being reduced
to point by a continuous deformation.
III. THE GEODESICS OF SIMPLY CONNECTED SPACES
11. If the Riemannian space 1s simply connected, there passes one and
only one geodesic between any tuo points. In fact we could assign to one of
these points the same role that was played by the point 0 1n the preceding
paragraphs. The other point N then actoilts normal co-ordinates determined
1n a unique way and which define the unique geodesic stemming from 0 and
abutting N. This geodesic could be extended to Infinity beyond the point N
(and thus beyond the first point as well); therefore it is never closed. The
relation 1n (10) shows that the geodesic OH is shorter than every other line
going from 0 to N.
The relation 1n (9) then shows that 1f 1 denotes the length of any
arbitrary line joining two points Η and H' with normal co-ordinates (хг) and
(хг ), we have
l £ (χ -χ ) + ... (χ -χ ) +... + (χ -χ )
By applying 1t to the geodesic ΜΓ of trlangl 0W', we can see that 1n any
geodesic triangle with sides а, Ь, с and angle.. А, В, С the Inequality
с2 Ъ a2 + b2 - Zab Cos С (11)
1s Obtained. In particular, in a right-angled geodesic triangle, the square
on the hypotenuse -Cs greater than or equal to the even of the equares of the
other too sides. The length of a geodesic 0A normal to a geodesic Г
passing through A thus Indicates the minimum distance from the point 0 to a
variable point of r.
12. The result of this last property 1s that the distance 0И of a
fixed point 0 to a variable point of a geodesic Г could not pass through
even a relative maximum; this 1s because the geodesic joining 0 to the point
of Г corresponding to this maximum would be normal to Г (see no. 95). On

346
note III
the other hand, as this distance 1s augmented Indefinitely when the point N
extends to Infinity on Г, we can see that 1t ecbilts one and only one ш1п1омя.
Consequently we oould drop one and only one perpendicular geodesic from a
given point 0 onto a geodesic Г. The length of this geodesic 1s the
distance from 0 to Г.
13. It 1s possible to recover this result 1n another way and
2
so, obtain some new results, by Introducing a new form of the da of the
space. Let us start with a basic geodesic Г, with an origin A of the arcs
on г and a sense of direction that 1s positive. At any point Ρ of г
with curvilinear abscissa u, let us elevate an arbitrary geodesic G normal
to Г. Let us attach to the point A a rectangular frame (R ) whose n-th
axis 1s tangent to Г. We shall attach to an arbitrary point Μ of G the
frame (R) which 1s derived from (R ) by parallel transport from A to Ρ
along r, then from Ρ to N along G.
The position of N will be defined:
1 By the abscissa и of the point P;
2° By the direction parameters (α,,...,α ,)
with respect to the frame attached to P.
3° By the quantity t й О whose product with
the length of the arc MP.
of the geodesic G at P,
Ύ
Φ
+ α
π-Ί
represents
Of the n+1 co-ordinates thus Introduced, the last η only Involve the
n- 1 products хг я ta..
On keeping to the notations of no. 3 we will have
' ω. Μ α .dt + ω.
it t
*J
ω..
a - ι «-υ
U,j ■ 1 η) ,
[12)
the forms ω
zero for
t-0,
da*,....da , and are all
ι π-1
ι , ω., being linear 1n du. uu,.
П i^ 1
save ω which takes the value du.
η
The structure equations,with the same conventions as above, lead to the
relations
IT
4
IF
*ij
и
da. + a
k"ki
**%,
R. . α ω
vjrs г в
U - 1.2,
,n-l)
[13)

note III
347
The Initial value of 3fe- (t < n) 1s thus da{, whilst that of -^ 1s
zero.
14. By the same procedure as 1n no. 4, we deduce fro» the relations 1n
(13) the Inequality
£<*♦* Φ-[(£)'*(W* ··*©')·
The same conclusions are deduced. The form Σώ. In <&,,...,<&„ ,, du ,
equal to f2u for t-0, and whose derivative with respect to t 1s zero
for t ■ 0, 1s for t ■ t 1s greater than dur, unless throughout the
Interval (0,t ) -rr- β °» ^еп 1" particular da. m 0.
37ш quadratia differential form Σ^>· ^n ^ da|,...,da , is thus
positive definite and great** than du .
As the de- of the space only depends on u and the products χ -ία.,
then to determine It, we could set t - 1 and a. ■ хг throughout. We can
straight away deduce that it is a positive definite quadratia form, greater
than du .
But It 1s possible to go about this another way and constrain the
parameters a, to the relation
7 7 7
We ascertain that we then have
de2 - dt2 + Щ + ... + ω2_} > dt2 + du2 . (15)
2
15. By starting with one of the above two forms of the de , we could
show, as In no. 9, that every point of the space 1s touched 1n one and one way
alone, by a geodesic G normal to Γ; this we already know. The Inequality
In (15) yields for us the theorem saying that In a right-angled triangle the
square of the hypotenuse 1s greater than the sum of the squares of the other
two sides.
Let us apply to the study of the geodesies of the space Lagrange's Equa-
tlons (see no. 37) serving us by means of the form In (15) of the de . The
equation relative to the parameter t gives us
Λ ι - ** .„

348
Note III
Now the function ω^ -^ which 1s zero for t ■ 0, has Its derivative with
respect to t constantly positive or zero. 1t 1s therefore positive or zero.
Consequently if it is displaced on a geodesic G of the space, the distance
t of a variable point of this geodesic, with curvilinear abscissa s, to the
basic geodesic Γ, hoe its second derivative with respect to в constantly
positive or zero.
The distance t could not then pass through any maximum. Two cases are
thus possible.
1° The distance t adnlts a minimum which, on account of a relation given In
no. 95, corresponds to a perpendicular geodesic Η μ common to the two
given geodesies (unless these two geodesies do not Intersect, In which
case the minimum of t would be zero). In this case the distance t from
a point N of G to Γ Increases Indefinitely when the point Η roves
away from M0, In one sense or another. The two geodesies acfcnU one and
only one common perpendicular.
2° The distance t 1s a constantly Increasing or constantly decreasing
function of β. It starts, for example, from +» for в - -« to tend towards
a limit fe, zero or positive, for в - +». We say that, 1n this case, the
geodesic G, oriented 1n the sense of the в Increasing, 1s asymptotic
to r.
16. If G 1s asymptotic to Г 1n the sense of the β Increasing, the
foot u of the perpendicular Η dropped from И onto Г, extends to
Infinity 1n a sense determined by r, then в tends to +«. Effectively, let A
be a fixed point of G and let A be the perpendicular dropped from A onto
Г. Let us take on the other side of A a point N on G; the distance N
decreases as в Increases and the angle AM 1s obtuse (see no. 95). Then on
account of (11), the distance A 1s greater than AH.
From the moment when AH 1s greater than A , the distance A stays
equally greater than A ; the point u then stays always on г on the same
side as a. As on one hand the distance A 1s augmented Indefinitely, the
point u tends to infinity on Г.*
*If G was not asymptotic to Г, this then would no longer be necessary.
Let us note as well that the point μ could not always be displaced 1n the
same sense, at least 1f n>2. In effect, let us elevate at a point Ρ of
Γ the different geodesies normal to Γ. They give rise to a geodesic hyper-
surface at Ρ , but which 1s not, 1n general, totally geodesic It 1s possible
then to find on this hypersurface two points N and N such that the geodesic
HN 1s not situated on the hypersurface. The foot of the perpendicular dropped
from a variable point on this geodesic on г will then pass through Ρ on
leaving 1t and thence returning to 1t.

Note III
349
On account of this, the distance from μ to G when μ tends to
Infinity, lithe sense 1n question, remains bounded (and less than h).
Consequently, if G is asymptotic to T, then aonvervely3 Г is asipwptotio to
G.
Similarly, 1t can be seen that two geodesies G, and G~ asymptotic* to
a third Г, in the вате sense as Г, are asymptotic to each other for 1t 1s
possible to find on G, and G? two points M. and M? going to Infinity
with the distance between them bounded.
Let us Include a theorem just as straightforward to prove, on account of
the fact that the distance to Г from a point И of G tends to zero when
Μ goes off to Infinity 1n the sense 1n question. If, In the domain of the
spatial points whose distance to Γ 1s less than a fixed number, moreover, as
small as we like, the R1emann1an curvature at any point and 1n any planar
direction has in its absolute value a nonzero lower bound; this is what occurs
1n a space with constant negative curvature.
Finally, 1t 1s possible to prove that through ewery point exterior to a
geodesic Γ, there passes one and only one geodesic asymptotic to Γ In a
given sense.*
IV. NON SIMPLY-CONNECTED NORMAL SPACES
17. If the given space ε 1s not simply connected, then the theorems
which came to be proved are not always true, but they are true for the
covering space ε', and the consideration of this covering space shows how they
must be modified. In particular, given two points 0 and A of ε, there
passes through these two points as many geodesies as there are 1n ε* from
points A* corresponding to A, that 1s to say as many irreducible pathe
joining 0 to A.
To see matters clearly, let us consider a point transformation (isome try)
о
of e' taking 0' to 0'. and leaving invariant the de of ε'. All of
these 1 sometries form a group β which could be called the connexion group
of ε, and which 1s similar to the holonomy group of a locally Euclidean
*A11 of these properties have been proved for n-2 by J. Hadamard 1n his fine
Memo1 re: Les surfaces a courbures ορροβέββ (Journal de Mathematlques pures et
appHquees, 5e series, t. 4, 1898, pp. 27-73). This eminent geometer has
furthermore Indicated the possibility of extending these to any n, by using the
theorem both stated and proved by him; 1n a space with everywhere negative or
zero R1emann1an curvature, the surfaces, rather than the geodesies, have their
R1enann1an curvature everywhere negative. See Sur la courbure done les β spaces,
plus de deux aimeneione (Procesverbaux des Seances de la Sod fete des Sciences
physiques et naturelies de Bordeaux, 1897-1898, pp. 85-87). For π ■ 3, this
property 1s an Inmedlate consequence of the theorem stated In no. 173.

350
Mote III
space (see no. 64). None of the non-1dent1cal operations of this group leave
Invariant a point of ε'; the group 6 1s therefore discontinuous.
It 1s possible by means of the operations of β to construct 1n the
space e' a fundamental domain ρ representing the space c. I.e., such
that to ewery point of с there corresponds one and only one point of p.
To simplify matters we shall assume n-3.
Let us consider the surface V. being the locus of points of e1 equl-
distant from 0* and from 0*. This surface evidently divides e' Into two
distinct regions. Each of these regions, for example, the one containing 0',
1s simply connected and homeomorphlc to Euclidean space. This 1s 1n keeping
with what could be obtained by constructing all the semi-geodesies stomalng
from 0* and on each of them taking a finitely determined segment 0'P' or
all of the sem1-geodes1c. It 1s quite clear that 1f the point P' 1s
equidistant from 0* and from 0'., then all the points M' of the geodesic 0'P'
situated between 0* and P' are nearer to 0* than 0' and all those which
are on the other side of P1 are nearer to 0* than 0*. The elementary
variation of the distance 0'M' 1s 1n fact equal to the displacement undergone by
the point M' when the point leaves 0', when the elementary variation of the
distance ΟίΜ' 1s equal to that same displacement multiplied by the cosine of
the angle which the geodesies 0'M' and 0^M* describe at M*.
We might add, though 1t 1s not absolutely Indispensable for what follows,
that the surface Ί\ 1s all of one piece. For 1f P' and Q' are two of Its
points and that all the points M' of the geodesic P'Q' that are situated
between P' and Q' are, for example, taken further from 0* than 0\t there
will be a point situated between 0* and M' on each geodesic 0'M' that
belongs to V.; this proves the existence of an arc of a curve joining P' to
Q' on the manifold V{.
18. The result of this 1s that the portion of the space ε* formed by
points situated on the seme side as 0' with respect to all the surfaces V.
constitutes the fundamental domain ρ required. This domain could be limited
by a finite or infinite number of faces 3, constituted by portions of
surfaces V. which correspond palrwlse, by generating isometries of the group Ш.
If
The domain ρ 1s simply connected and homeomorphlc to Euclidean space, since
1t could be obtained by taking on each sem1-geodes1c stemming from 0* a fixed
segment 0'P' (or the sem1-geodes1c completely).
On account of this, we can see that if an n-dimeneional manifold is вив-
oeptible to booing a normal metrio with aurvature that is everywhere negative
or aero, it must then poeeeee the property that onoe made simply oonneoted by
a partitioning in (n-1) dimensions, it beoomee homeomoprhio to Euclidean epaoe.
19. We could finally generalise a comment made 1n (no. 68) for locally
Euclidean spaces; 1t 1s that the connexion group б contains an infinity of

Bote III
351
operations. In fact, 1f the group б was finite, the point 0' would have
a finite nunber of homologues Oj.OI 0£ ,. Let us denote by :»{ the
distance between an arbitrary point of ε' to the point 0'. and consider the
2 2 2 г 2
function r + r, + ... + r. ,. It evidently aotalts a lower bound a ,
attained at least at one point A'. The extremal condition at A* requires
that 1f we denote by a£ the angle described at A* by the geodesic A'O'
with any fixed direction, we have (see no. 95).
Let
bo;
A'B'.
V
В then.
and a.
cos α + r, cos a-, + ...
о 1 1
be any point 1n ε', d
the angle described at
We have on account of (11),
4
ъ r^ + cf - 2ifr£cosa{
+ Vi cos V
the distance
i"° ■
A'B', r\
A' by the two geodesies
the distance
A'O.
ι
and
and consequently
2 2
The lower bound a of the function £r. could only then be attained at a
i
single point A'. It 1s then clear that ewry operation of the group б (the
points 0',o;,...,0! being Interchangeable) would leave Invariant the point
A', which 1s not possible.
V. CLOSED 6E0DESICS IN NORMAL, NON-SIMPLY CONNECTED RIEMANNIAN SPACES
20. Let us consider 1n the simply connected covering space ε' of the
space ε the fundamental domain ρ and one of the homologous domains p{
resulting from p by one of the displacements of the connexion group 6. To
the curves joining an arbitrary point H' of the domain ρ {boundary included)
1n ε', to a homologous point H'. of P., there corresponds 1n the space ε
some closed curves emanating from an arbitrary point Η and returning to 1t.
All these curves form a class (C.) of closed curves reducible to each
other by a continuous deformation, since the domain ρ 1s connected as well
as p{. Let us consider 1n particular those curves which are geodesies.
Through a point Η of the space ε, there passes one and only one which has
for Its Image 1n ε1 the geodesic joining H' to H1..
In general the two semi-tangents at Η to the geodesic 1n question form
an angle that 1s not a flat angle. In other words, the geodesic does not
Impinge upon Itself when 1t returns to the point H. We say that the geodesic

352
Rote III
1s closed when Its two semi-tangents at Η are opposite. We are going to
prove the following theorem:
Theorem. The враав ε admits at least one closed geodesic belonging to the
class C.. It adnits at least boo if the fundamental domain Ρ is bounded,
21. Let us start with the following note, a consequence of the formula
1n no. 95, relating to the variation of a geodesic arc. Let us consider the
length l of the geodesic of the class (C{) starting from Η and returning
to 1t, and let us take a point N 1nf1n1tes1mally near to H. The principal
part of the variation undergone by the length 1 1n passing from Η to N
1s equal to
-MN(cos α + cos β).
where α and β denote the angles described by MN with the two semi-tangents
at Η to the first geodesic. If this geodesic 1s closed, then this principal
part 1s zero; 1f 1t 1s not, then 1t 1s possible to find points Η as close as
we like to Η such that the angles α and β are acute, and also, points N
for which these two angles are obtuse.
Consequently, the closed geodesic is characterised by the property that in
the space ε", the distance M'M', being a function of the variable point H*
of the domain P, is stationary.
It will suffice then to find a point H' of ρ such that the distance
M'M' 1s a relative max 1 nut) or minimum 1n order to obtain a closed geodesic of
the class (C.) 1n the space с Now the distance H'H'. 1s a continuous
function of the point M' 1n the closed domain p. This function 1s never
zero; 1t has then a lower bound which 1s attained, thus proving the theorem.
If the domain ρ is bounded, the distance HM' is also bounded; 1f then a
maximum which 1s attained, whence the existence of a seoond oloaed geodesic.
Finally, it could be that this distance 1s constant, in which case there
passes through each point Η of the space ε, a closed g deslc of the class
(C). This occurs 1n the two-dimensional R1emann1an space, representative of
the curves of genus 1 endowed with a R1emann1an (Euclidean) metric, as
Indicated 1n no. 71.
22. Let us consider 1n the space ε a closed geodesic and 1n the space
ε' Its Image ΗΉ1; let us extend 1t Indefinitely 1n this space. The d1s-
placement of 6 which takes H' to H\ take the geodesic arc M'M' to Its
extension H'.H'., then the second arc to Its extension M^MJ, and so on.
Through every point P' of 0' there passes a geodesic asymptotic to the
geodesic I" which we are going to construct. Thus there results the
following theorem:

Bote III
353
Theorem. Through every point Ρ of the βραοβ ε there paeeee a geodeeio G
авущ>ЬоЫа to a given olomed geodeeio Г.
To say that G 1s asymptotic to Г 1s to say that the distance from
G to Г tends towards a finite limit and the foot of the perpendicular
dropped from a point of G onto Г runs across the closed geodesic an
Infinite number of times. The distance from G to Г tends towards zero of the
R1emann1an curvature at the different points of Г and following the different
planar directions at these points,* 1s never zero.
A simple example 1s provided by the hyperbolold with one sheet and the
closed geodesic constituted by the ellipse of Its "throat."
*In practice 1t suffices to consider tangent plant directions to the geodesic.

NOTE IV
THE GEODESICS OF NORMAL RIEHANNIAN SPACES
I. AN EXISTENCE THEOREM
1. In Note III ме proved the theorem following which any two points of
a normal R1emann1an space with everywhere negative or zero curvature could be
joined by a geodesic. The proof explicitly utilises the hypothesis relating
to the sign of the R1emann1an curvature. This theorem 1s nevertheless true
for every normal space.
We shall вике the same analytic hypothesis as 1n Note III. We are first
going to prove the following preliminary theorem whose bearing 1s purely local.
Theorem I. Given a point 0 of a Riemomnian араоея let Ση be the hypersphere
Iooub of points Μ of the apace whose distance OH to the point 0 is less
than or equal to R. There exists a nwber R possessing the following
property: there exists one and only one geodesic arc joining 0 to an arbitrary
point Μ of Ση without leaving Σ»* and *Ь& aro has for its length the
distance OM.
Let us recall what we have named the distance between two points AB of
a normal R1emann1an space, and denote by AB, the lower bound of the lengths
of arcs of rectlflable curves joining these two points. More exactly, we
assume that on these arcs, the co-ordinates of a moving point are continuously
dlfferentlable functions of a parameter.
2. With this established, let us consider all the semi-geodesies stemming
from 0 and take on each of them, starting from the point 0, an arc of
length R. These geodesic arcs form a domain ZL. Let us now Introduce the
normal co-ord1antes χ relative to a rectangular frame of origin 0. The
co-ordinates иг which occur 1n the given fundamental form of the space are
functions χ admitting continuous partial derivatives of the first two orders,
and their functional determinant with respect to the χ , 1s non-zero at 0
and will be non-zero within a sufficiently small neighbourhood of 0. We may
assume, by replacing the number R by a smaller number as the case may be,
that the functional determinant 1s non-zero for the domain ZL defined by
(x1)2 + (x2)2 + ... + (xn)2 * R2 .
This domain 1s represented 1n normal Euclidean space with origin 0 and
rectangular co-ordinates χ , by the hypersphere S„ centred at 0 and with
radius R.
355

356
/tote IV
We are firstly going to prove that every point of ZL could be tied to
the point 0 by one and only one geodesic of length less than or equal to R,
otherwise there would effectively exist 1n the hypersphere S„, two distinct
points (x) and (χ1) corresponding to the same point u of DR. Now we have
by putting
ξ* -χ* + θ[(χ*)' - χ*] (0 < θ < 1) .
The points (χ) and (χ1) being 1n the hypersphere SR, the point (ζ) lying
on the straight line joining these two points is also 1n this hypersphere.
If the two points χ and x1 correspond to the same point и of 0-, the η
differences (x )' - χ satisfy a system of л linear, homogeneous equations
1n η unknowns, the determinant of the coefficients of the unknowns being
nonzero; the two points 1n question could not therefore be distinct. There te thus
a bijeotive oorreepondence between the pointe of S„ and the points of ZL.
3. We are now going to prove that the unique geodesic arc joining 0 to
a point Μ of DR of length at most R has its length equal to the distance
ON. Once this has been done, 1t will result that every point Ρ 1η the
exterior of iL could not belong to Σο* since every curve arc joining 0 to Ρ
must clear the boundary of 9g. If Q 1s the first point of the curve arc on
which 1t occurs (the length of the curve arc OQ 1s already at least equal to
R), such that the curve arc OP 1n question, 1s always greater than R and
the point Ρ could not then belong to Σο· The domain ZL 1s Identical to
Σρ and the Immediate result 1s the theorem stated 1n no. 1. To prove that
the geodesic arc joining 0 to a point Μ of Σα has Its length equal to
the distance OM, let us Introduce co-ordinates t, α,, α-,.,-.,α employed
1n the theory of normal co-ordinates, by assuming that
a? + a? + ... + a2 - 1
1 2 η
The equations 1n (4) of Note III Imply
h {аг h]' eA ■ ° · ai h
from which 1s deduced, on account of the equations 1n (3) of the same Note,
2 2 2 2
l с η
9 -9 -2
■ dtc + ωί + ... + ω'
ι η

Note IV
357
and then 1n particular,
daZ > dtz . (1)
Let then Η be a point of ZL. The value of t at this point 1s equal to
the length 1 of the arc of the unique geodesic, length less than or equal to
R, that connects 0 to M. Now let us join 0 to Η by a rectif1able curve
С not leaving ZL·; at each point of this curve arc, t has a suitably
determined value between 0 and 4. The Inequality in (1) gives for the length λ
of this curve arc, the Inequality
λ > I \dt\ > J df I ,
and the length λ of this curve arc 1s thus greater than the length l of the
geodesic arc. For all the more reason, 1t 1s still the same 1f 0 1s joined
to Η by a curve arc leaving flu, to eventually return there.
The length 1 of the geodesic arc 1s thus « well defined lower bound of
the lengths of the curve arcs joining 0 to H.
Q.E.D.
4. The proof as given leads 1n a sense to « somewhat more general theorem
which 1s as follows.
Theorem 11. Given in a Uiemannian spaoe a domain 9 and a point 0 in the
interior of this domain, ы* авешгш that «v«ry point Η in th* interior of this
domain could be joined to the point 0 by one and only one geodesic arc not
leaving the domain. The length of this geodeeio arc ie leea than the length
of every other curve ото joining 0 to Μ without leaving the danain.
It suffices to Introduce Into the domain normal co-ordinates with origin
0 by means of the co-ordinates t, a*ta09..*ta indicated in no. 3. Each
point Η therefore corresponds to a unique system of co-ordinates t, α-,,α?*
...,α , and the Inequality 1n [1) remains true tnd sufficient for the proof
of the theorem. In particular the result 1s that the geodesic joining 0 to
Η without leaving the domain realises a relative minimum distance, unless
1t 1s naturally possible to affirm that this minimum 1s absolute.
5. We are now going to move on to the fundamental theorem that we had 1n
mind.
Theorem III. In a normal Riemannian space, there exists at least one geodesic
joining any Ux> given points M,N and this geodesic has for its length the
distance HN.

358
ПоЫ IV
Let us recall that 1n a normal space, every Infinite bounded set of points
actarits at least one limit point (see no. 56). Let us denote by 1 the
distance |MN| between two given points. There exists an Infinite faelly of rec-
tlflable curves joining Η to N and whose total length tends to 1. Each
of these, C{ say, has a centre point ?_. such that on the line C{, the
length of the arc MP. equals that of the arc P.N. Forming a bounded set,
all of these points actarit at least one limit point P. The distance |MP| 1s
at most equal to 4, since there exists a collection Of rectlflable curve arcs
joining Η to Ρ and whose length tends to ΐ > similarly for the distance
|PN|. But as we have
IMPI + IPNI i IMNI - I ,
this 1s only possible 1f
IMPI
|pn| ■£
We have thus proved the existence of at least a point Ρ whose distance to
each of the points Μ and N 1s equal to half the distance HN. He shall
denote the point Ρ by the notation M,/2 and the points Μ and N could
be denoted by Mn and M, respectively. Similarly, there exists a point
t
M,/4 situated at a distance -ξ from Μ and M^, *nd * point M^ at a
distance 4 from Μ,,- β™1 fro" *\- Tne distance Ι^ι/Λ/λΙ 1* e°.u*l to
4 ; this results from the Inequalities
1^/^3/4! * lMi/4Mi/2l + lMi/2M3/4l Ί
and
Ι^Λ^Ι * l"i/4pl3/4l * Ι^ΛΙ
2 1
where
''Ь/Лм! *f
31
Similarly the distance |M0M3/4I 1s equal to ^ as this results from the
Inequalities
l"oM3/4l s IVW + Ι^/Λ/αΙ ■ τ*
and
'MoM3/4l + lH3/4Hll * 1ИЛ1 "l
where lM0M3/4l kj. we again have IH^Mj | ■ Ц- . This can be continued.

ПоЫ IV
359
Given an arbitrary Integer и, there exists 2я-1 points Κ,/φΐ (ρ ■
1,2,...,Z"-1) such that the distance between any two of them, with Indices
·£.·*:<?>*>. 1s equal to *=& I.
2й г" 2я
If now α 1s a number taken between 0 and 1 that 1s not of the form -ζ- ,
2я
we could regard 1t as the limit of an Infinite collection of numbers
Pi ?2 Pn
2 2* 2я
Being bounded, the set of corresponding points aomlts at least one Unit point
P. This Unit point 1s unique, for 1f there did exist another P', the
distance |PP'I would be less than the sum of two numbers arbitrarily snail, on
account of the triangle Inequality
|PP'| s |PQ| + IQP'I .
We can denote the Unit point Ρ by the notation Η , and easily prove that
the distance from Η to any point Μ -2- is equal to (a - -2-) t and then
we have 1n general
ΜαΜβ - (β-α)Ι (β > α) .
6. The above considerations thus lead us to the conclusion that there
exists a curve joining Μ to Ht uhoee moving point ie a continuous function
of a parameter t varying between 0 and 1, euoh that the diatanoe betaeen
изо points on this move corresponding to the parameters t, tr (t* > t) ie
equal to (tr - til . This is a Jordan curve which we shall call Г.
It remains to prove that this curve 1s a geodesic, consequently admitting
at each point a continuously variable tangent. Let us take on the Jordan
curve a particular point Ρ with parameter t . There exists a hypersρhere
Σ (R) of centre Ρ and radius R having the properties stated 1n Theorem I.
The point Q of Γ, with parameter t + R, Is on the boundary of this hyper-
sphere and the geodesic ace PQ of length R joining Ρ to Q, an arc that
we shall denote by G, realises the distance PQ ■ R. We are going to prove
that the arc QP of Γ is identical to the geodesic arc G. Effectively let
Ση(Κ') be a hypersphere with centre Q and radius R, possessing the
properties stated 1n theorem I, and let us take between Ρ and Q on Γ a point
Η whose parameter 1s greater than t + R - R\ This point Η 1s at once 1n
the Interior of Σ W and £q(R')- Therefore there exists a geodesic arc
of length QH joining Q to H. As the sum of the lengths of these two curve
arcs 1s equal to the length PQ, 1t happens that the point Η 1s on the arc G;

360
Hote IV
otherwise the length of the line formed by the two geodesic arcs PH and HQt
a line entirely within Σρ(Ό» w°uld be greater than PQ. If R' 2 R. the
argument applies to all points on the arc QP of Γ, which so becomes the
geodesic arc G. If R' < R, the results 1s only true for the arc QQ. of
Γ, Q1 being the point with parameter t + R - R'. He are going to show that
the property 1s nevertheless true for all arcs QP of Γ.
Effectively let t + α (0 < α < R) be the upper bound of the set of
parameters of the points of the arc QP of Γ which are not on the arc G.
The point К of Γ of parameter t+α 1s Itself on this arc G since 1t
could be regarded as the Hm1t of an Infinite collection of points of Γ whose
parameter constantly decreases to tend to t + a; now as all these points are
on G, the Unit point К 1s also on G. But then 1t suffices to re-Iterate
for к the above same argument, and we arrive at a contradiction.
Q.E.D.
7. On leaving the point Η with origin Γ» there will then exist an
arc MP. of Γ which will be a geodesic arc, and then an arc Ρ,Ρ- of Γ
which will be another geodesic arc. But let us now note that thie second are
will be the extension of the firmti otherwise, on rounding the angular point
produced at P., we would actually construct without any difficulty, a curve
arc joining Μ to P. and which would be less than the distance HP..
We easily deduce that every Jordan curve 1s one and the same geodesic arc.
It suffices to argue Identically as we did In no. 6. Theorem II 1s thus
completely proved.
Note. Theorem II might collapse for R1emann1an spaces with an Indefinite
fundamental form and presenting all the characteristics of regularity that we
might wish for.*
II. THE FOCAL HYPERSURFACE OF A POINT
8. Let us consider a point 0 1n the space and Introduce a system of
normal co-ordinates xb having this point as the origin. Let us take a semi-
geodesic leaving 0. In what follows, then η forms ω£ {x,dx) will start
by being linearly Independent; there might be an occasion where they cease to
be so. Let A be the first point where this happnes. In order to keep to the
general case, let us assume that at the point A the matrix of coefficients
of ω1 ydx2,... tdxn 1n the forms ω. 1s of rank n-1. There will exist at
*See in E. Cartan, Leoone aur la gtomStrie projective complexe (Paris,
Gauth1er-V1liars, 1931, pp. 1B3-184), the example of such a space of dimension
15, having the property that 1t 1s possible to find a couple of points through
which there passes an Infinity of geodesies, a couple of points through which
there passes one and only one geodesic, and a pair of points through which
there does not pass any geodesic.

note IV
361
A coefficients α,,α»,...,α, not all zero, such that 1f a displacement from
A 1s made by giving the dx1 arbitrary values, we have
α.ω. + α^ + ··· + <у«»я ■ 0 . (2)
In particular, this relation will occur 1f a displacement 1s roede on the
geodesic OA; the geodesic 1s thus tangent to the (n-1) dimensional planar
element which has cu,a2,...,a as Its covarlant components. Now these
components are those of the normal vector at A to the hypersurface Φ obtained
by equating to zero the determinant of the coefficients of the dx1 1n the
forms ω.. He thus arrive at the following theorem:
If by в bating that the linear forms ω· in dx , dx ,...,dx are not
independent, the resulting equation adnits real solutions, then they define a
hypereurfaoe tangent, at each of its points, to a geodesia stemming from 0.
This hypersurface 1s called the fooal hypereurfaoe of the point 0 and
the first point where a geodesic stemming from 0 touches this hypersurface
1s called the foaue of the point 0 on the sem1-geodes1c 1n question.
9. Conversely, let us assume that there exists a hypersurface Σ to
which all the geodesies stemming from 0 are tangents, at least 1n a certain
neighbourhood of a particular geodesic. Let A be a point of Σ situated
on a geodesic OA tangent to Σ at A and let В be a point of the space
1nf1n1tes1mally near to A. The vector AB is the geometric sum of an
Infinitesimal vector joining the point A to the point B' of the geodesic OB
where this geodesic touches Σ and of an Infinitesimal vector B'B. The
vector AB 1s therefore tangent to Σ at A. Consequently 1n displacing,
the departure from A 1s Immaterial, that is, by giving the dx arbitrary
values, there exists a finite relation between the η forms ω. of the type
1n (2). The point A as well as all the points of Σ thus belong to the
hypersurface defined 1n no. B.
10. Let us consider at each point A of Σ the unit vector to which
the sem1-geodes1c stemming from 0 is tangential; we thus obtain a tangent
vector field on Σ. Let us consider a trajectory curve of this vector field
and let A be a point on this trajectory. For the general case, let us assume
that A 1s an ordinary point of Γ directed 1n the sense of going from A to
0 along the geodesic OA. Let В be a neighbouring point of A on this arc.
With reference to Darboux* we are going to prove that the arc length of
a geodesic OA cannot be a relative minimum for the lengths of the curve arcs
inftnitesvnalty near to a geodesic arc joining 0 to A.
*fheorie das surfaces, t.III, pp. 86-88.

362
Mote IV
In fact because of the relation that gives the variation of a geodesic arc when
Its extremities vary, It can be seen that In going fro» В to A, on the arc
In question, the goedeslc arc length has undergone a variation equal to the arc
length of BA of the trajectory* of Σ passing through A and B:
arc geod. OA ■ arc geod. 08 + arc BA
Now the trajectory is not a geodesic, for through the point A there
cannot pass two distinct geodesies that are tangent to each other. Therefore
there exists a curve arc joining A to В smaller than the arc of the
trajectory and consequently there exists a path from 0 to A smaller than the arc
of the geodesic OA. The proposition Is thus proved.
We refer the reader to the text of Darboux for the very straightforward
proof of the theorem on account of which, the length OH of the geodesic in
question ceases to be an absolute minimen distance before attaining the fooue
of the point 0, at least in keeping iHth the above hypothesis.
11. We have assumed that when tha determinant of tha matrix of
coefficients of the dx of the forms шк Is zero, the rank of this matrix Is
n-1. It could happen that this rank reduces to n-h (h>l), for example.
In place of a hypersurface Σ* ** could have an (n-h) dimensional
manifold, a focal manifold of the point 0 which at each of Its points would be
tangent to an Infinity of geodesies stemming from 0 depending on h- 1
parameters. The results stated for the general case are still valid.
2
An extreme case Is where we might have я-л β 1. In this case the as
of the space would be reduced to a perfect square at each point of the focal
manifold. By Introducing the co-ordinates t3 а·,, а~, -·-, a with
7 7 7
a\ + a% + ... + a ■ I ,
12 η
we have
,2 ,J . -2 . -2 . . -2
as * at + ω. + ω0 + ... + ω
12 η
о
If the right hand side Is a perfect square then it can only be dt . The result
Is that If we pass from a geodesic OA (A being the focus of 0) to a
geodesic Inflnlteslmally near and if the length OA Is taken on this geodesic·
tha extremities A and A' of these two geodesic arcs are at a zero distance.
Consequently, the focal manifold reduces to a point and all the geodesies
stemming from 0 are going to ραββ through a fixed point.
This Is what happens for spaces of constant positive curvature. The focal
manifold Is the locus of points situated on each geodesic stemming from 0 at
tha distance — [see formula [15) In no. 221]; all of these points become a

Vote IV
363
single point. In this case, each geodesic arc of length less than — realises
an absolute minimum of the distance, at leaet if the epaoe is eimply oonneoted
(spherical spaces), for In elliptic space, the distance between two points
never exceeds -Ϊ- .
2Д
12. Let us end with the statement of a closing theorem which Is a
consequence of theorem III and of the preceding considerations.
Theorem IV. If in a normal epaoe, all the emi-geodesiae etemting from a given
point 0 are eaoh taken to the focus of the point 0 (авешгСпд there ie one),
then all the points of the epaoe are attained at leaet onoe-
In the case of a spherical space, all the points are attained once and only
once (with the exception of the antlpode of 0 which Is attained an Infinity
of times). In the case of elliptic space, each point Is attained twice;'the
antlpode of 0 here becomes the point 0 Itself. In general there will be
an Infinity of points attained several times, even If the space Is simply
connected.

NOTE V
COMPLETELY INTEGRABLE PFAFFIAH SYSTEMS
1. Let us consider a system of total differential equations
e1 - о , o2 - о , or - о , (l)
where the θ denote r linear differential forms constructed with n + r
variables ■с1|*2,...,*"+г and their differentials, the coefficients admitting
continuous first order partial derivatives. We shall assume that these forms
are linearly Independent; putting It another way, we shall assume that the rank
of the matrix Μ formed by the coefficients Is r. We shall use geometric
terminology on regarding the χ as the co-ordinates of a point In n + r
dimensional space and say that a point A of this space Is generic If for the
co-ordinates of this point, the rank of the matrix Я Is exactly equal to r.
Let us consider In the space a continuously dlfferentlable n-dlmenslonal
manifold V, I.e. such that the co-ordinates of one of Its points can be
expressed In terms of continuously dlfferentlable functions of η parameters.
We shall say that It Is an integral manifold of the system In (1), or that it
constitutes a solution of the system In (1), If each of Its linear tangent
elements, defined by the co-ordinates of a point of V and the direction
parameters dx\dx ,...,dzn+r of a tangent to V at this point, satisfy the
equations In (Ί).
If the point A In question Is generic and if the determinant formed by
the last ρ columns of the matrix Is nonzero at this point, which could always
be assumed, the equations In (1) could be solved with respect to dx ,....
ώΡ** In a neighbourhood of A. The result Is that xn+1,... .x"*2*, considered
12 η
as functions χ ,«,...,« on V, admit continuous first order partial
derivatives at A and at each point of V sufficiently near to A. By putting
χ
■ tT (a ■ l,2,...,r)
to simplify matters, and assuming that the Index t takes the values 1,2....,
n, the manifold could be defined by г equations
βα- βα(χν,...,χη) (a-1,2 r) , (2)
where the sa admit continuous first order partial derivatives. We could say
that every point of the Integral manifold that Is a generic point of the space
Is a regular point of the manifold. This will express the possibility of an
analytical representation of V as In (2) In the neighbourhood of this point.
365

Збб
Hot* 7
2. Definition. The Pfaffian eyetm in (1) is said to he ocmplsUly inte-
grable if through every generic point of the epaoe, there раввее an integral
manifold.
Naturally enough, we only require the existence of this Integral manifold
In a sufficiently small neighbourhood of the generic point In question. It is
easy to find the necessary conditions for complete Integrablllty. Effectively,
let us for* the exterior differentials of the fonts θ ; these are exterior
1c a
quadratic fonts constructed by the variables χ and their differentials. If
we now consider the neighbourhood of a generic point A for which the above
hypothesis Is true, the for» db could be expressed In terms of the η dlf-
i α
ferentlals dx and the ρ forms ββ which are linearly Independent. We will
then write
d\ ' 7 ^.I***^ + W^V + 7 Si* [Vy] · (3)
where the summation Indices i and j vary between 1 and л, and the
summation Indices vary between 1 and 2*; we can however assume that
Let V then be an Integrel manifold containing the generic point A. On
displacement on this manifold, the fonts θ are Identically zero and thus It Is
the same for the forms dQ . In the neighbourhood of A, for which the analy-
1 2 и
tlcal representation In (2) Is valid, the variables x\x χ remain
Independent and consequently the coefficients A.. of the relations In (3) are
zero. Therefore In such a neighbourhood the existence of Identities of the form
ла - g£ eB] (4)
comes about, where the ω ^ denote conveniently selected linear differential
а
forms whose coefficients are continuous functions. It Is possible to take as
an example
4/■B? *f ♦ KY\ · (s)
we could also write the relations In (4) more simply In the congruence form
dea ξ о (mod βΓβ2 er)
With this established we are going to prove the foil owlnq theorem:
Theorem. Гп оМл* that the Pfaffian ayatmm ίη (1) im to be aornpletely inte-
groble, it ie neoeeeary and euffioient that in a neighbourhood of every generic

ItoU V
367
with origin A Into the n-dlmenslonal space of the χ . It suffices to put
point of the space, the exterior differentiate dQ of the left hand eidee of
the system are congruent to zero (nod θ.,θ0 θ ).
I С Г
3. The condition Is necessary: this results from what came before. We
are going to show that It Is sufficient. Let A be a generic point of the
space satisfying the above hypothesis. Every Integral manifold passing through
A admits an analytical representation of the type In (2). Let (χτ) , (βα)
be co-ordinates of A. We are going to Introduce a system of polar co-ordinates
In A Into the n-
x* - (x\ - t β* .
where t will be a new co-ordinate that Is essentially positive. For each
system of values of the a*, assumed to be not all zero, the point (x£), as t
varies, describes a half-life stemming from A and all of these half-lines
fill out the entire space. The aa then become continuously dlfferentlable
12 η
functions of the new n+1 super-abundant co-ordinates α ,α: ,...,α , t.
We are firstly going to fix the co-ordinates а\аг an whilst varying
t. The aa then become functions of t defined by a system of ordinary
differential equations obtained by replacing, In the equations In (1). the dx by
a'dt and dx1** by ώα. It could be written as
4f ■ .°(Л*,.В) . (б)
This system admits a well-determined solution corresponding to the Initial
conditions «a - (ea) for t ■ 0. This solution exists In a certain Interval
λ о
(0,t ) of the Independent variable t; as for t It could depend on the
к ° t °
a , but If we assume for example that the a are constrained to satisfy the
relation (a1)2 + (a2)2 + ... + (an)2 - 1, this number t will have a
nonzero lower bound that could still be called t . The result of this Is the
о
existence of functions a defined In a sufficiently small neighbourhood of
the point (χ£) , bound by the Inequality
[x1- (*\]2 + ... +[*"- (*\]2< t\
With this established. If we calculate the differentials of the functions
aa of tj a , a2,..., an so obtained and were to likewise replace the dx1
by ατ&'+ t dat . the forms θ could be written as
θ - Η .da
α on.
α
t
Following the way the functions aa had been determined, the term In dt of
θ Is In fact Identically zero. We could add the fundamental remark that for
t-0, the functions Η .(t.a) are identically sera. In fact for t-0, that

368
Note V
Is, at the point A, the co-ordinates χ and «a keep the fixed values
(χτ) and (sa) when the co-ordinates a ta%...tan are varied so that
о
their derivatives with respect to these co-ordinates are all zero at A.
Let us now apply Identity (4) that Is true In the neighbourhood of A;
let
ω* - pf(t.a)A + Q* (Ьла)аа1 .
at
By only preserving the terms containing dt In this Identity (4)· we will
obtain
whence for each value of the Index t, the relations
ТГ " >Xi <7>
The functions H,.,H2.,...,H. thus satisfy a system of linear homogeneous
differential equations whose coefficients Ρ are continuous functions of t,
with the Initial conditions Η . - 0 for t-O. Consequently these functions
are Identically zero; It Is therefore the same for the left hand sides of the
system In (1). There exists then an Integral manifold passing through the
point A. We can see moreover that this manifold could be obtained by
Integrating a system of ordinary differential equations, I.e., the system In (6).
4. Remark I. We could assume that rather than consider the ordinary
(г+я) dimensional space· we consider a manifold according to the regions
necessitating the systems of variables co-ordinates. Nothing would change In
the theorem and there would only be the standard modifications In dealing with
the proof.
Remark II. The proof given In no. 3 shows that for a given Pfafflan system
In (1), even one not completely Itegrable, It Is only possible that one Integral
manifold passes through a generic point of the space, for If such a manifold
exists. It Is given by the Integration of the differential system 1n (6).
Remark III. Even for a completely Integrable system. Identities such as
(4) would cease to be effective at а попдеплНа point of the space. A very
simple example Is provided by the equation
θ = xdy - ydx ■ 0
which Is evidently completely Integrable, as Is every ordinary differential

Note V
369
equation. A relation
d& - [(Adc + Βφ)θ]
or by calculating,
2[dz <fc] - [(Adr + My){xdy - ydx)1
I.e.
2 - Ax + By .
Is only possible with functions A and В finite and continuous at the point
(x*y) If χ and у are not both zero, that Is, If the point Is generic.
We could however replace the relations In (4) by the relations
[e^g ... 6rdea] - 0 (a - 1,2 r) (B)
which are equivalent to the* at every generic point of the space. They do not
cease to be exact at the nongenerlc points by virtue of continuity; every non-
generic point could be regarded as the limit of an Infinite family of generic
points.
The equating to zero of the г exterior forms of degree r + 2, [Θ.Θ. ...
θ άθ ], thus expresses directly the conditions for complete Integrablllty of
the system In (1).

BIBLIOGRAPHY
Gauss. Olsqulsltlones generales circa superficies curves (present* a la
soclete des Sciences de GBttlngen, 1827; puclle Comment Soc. G5tt. recent.,
6, 1023-1827; Sesawa. Werke. 4, Gottlngen, 1827, p. 217 sulv.; trad, fr.,
Novelles Annales Hath., 1652).
Rlemann, B. Ueber die Hypothesen we 1сhe der Geonetrle zu Grunde Hegen
(Habllltatlonschrlft 1854; GCtt. Abh.. 13, 1868, p. 1; Gesaw. Werke. 2e
ed.f Leipzig, 1892. p. 272).
This fundamental work has been edited, with comments, by
H. Weyl (Berlin, 1919, 3e edit.. 1923).
Rlemann, B. Commentatlo mathenatlca, etc. (Gesam. Hath. Werke, p. 370-
380).
Lane, G. ίβςοηε sur les coordonnees curvlllgnes (Paris, 1B59).
Chris toff el, Ε. Β. Ueber die Transforation der homgenen Differential-
ausdrUcke zwelten Grades (J. de Crelle. 70, 1869, p. 46-70).
Beltrami, E. Sulle d1 Interpretazlone del la Geonetrle non euclldea
(61omale d1 Hatem., 6, 1869, p. 284-312; trad. J. HoUel. Ann. Ecole Norm..
6. 1869. p. 251-2ββ; Opera Hat., I. Milan, 1902, p. 371-465)"
Beltrami. E. Teorla fondamentale degll spazll d1 curvature costante
(Annall dl Haten.. 2e serle, 1868, p. 232-255; trad. J. HoDel. Ann. Ecole
Norm., 6, 1869. p. 345-375; Opere Hat.. I, p. 406-4297"!
Schur. F. Ueber den Zusamnenhang der RSume constanten KrUmmungsmasses mlt
den projektlven RSumen (Hath. Ann., 27, 1886, p. 537-567).
Klein, F. Ueber die sogenannte nlcht-euklldlsche Geonetrle (Hath. Ann..
4, 1B71, p. 573-625; 6, 1B73, p. 112-145).
Klein, F. Zur nlcht-euklldlschen Geonetrle (Hath. Ann.. 37, 1890, p. 544-
572).
Klein, F. Nlcht-euklldlsche Geonetrle (lecons autog., 2 vol. GOttlngen,
1B93).
Klein, F. Conferences sur les Hathenatlques. trad. L. Laugel, (Paris,
Hermann, 1B9B).
Killing, U. Die nlcht-euklldlschen Raunformen In analytlscher Behandlung
(Leipzig, 1885).
Killing, U. Ueber die Cllfford-Kleln'schen Raunformen (Hath. Ann., 39,
1B91, p. 257-27B).
Darboux, G. Lemons sur la theorle des surfaces, t. Ill (Paris, Gauthler-
Vlllars, 1B94).
bis. Cotton, E. Sur les varlfttes a trols dimensions (Ann. Fac. Sc.
Toulouse, 2e serle, 1, 1889, p. 385-438).
Rlccl, G. Formole fondamentall nella teorla generale delle varleta e della
loro curvature (Rend. Accad. Llncel, 5e serle, 11, 1902, p. 355-362).
371

372 Bibliography
IB. Rlccl, G. Sulle superflcle geodetlche In una varleta* qualunque e In
partlcolare nelle varleta a tre dlmenslonl (Rend. Accad. Llncel. 5e
serle, 12, 1903, p. 409-420).
19. Fublnl, G. Sugll spazll che ammettono un gruppo continue dl novlmentl
(Ann. d1 Hat.. 3e se>1e, B, 1903, p. 39-81).
20. Rlccl, G. Dlrezlonl e Invarlantl prlnclpall dl una varleta qualunque
(Attl Ri Istlt. Veneto. 63, 1904, p. 1233-1239).
21. Rlccl, G. Sulla detemlnazlone dl varleta dotate dl propriety Intrlnsche
(Rend. Accad. Llncel. 5e seYle, 19, 1910, p. 1B1-1B7).
22. Levl-Clvlta. T. Nozlone dl parallellsno In una varleta qualunque (Rend.
Clrc. mate». Palenoo. 42, 1917, p. 173-205).
23. Sever1, F. Sulla curvature delle superflcle e varleta (Rend. Clrc. mat.
Palenoo. 42, 1917, p. 227-259).
24. Bomplanl, E. Sugll spazl curvl (Attl R. Istlt. Veneto» 80, 1921, p. 355-
386, B39-B59, 1113-1145).
25. Wevl, H. Space, Time, natter Dover
26. Strulk, D. J. GrundzUge der mehrdlmenslonalen Dlfferentlalgeonetrle In
direkter Darstellunq (Berlin, J. Springer, 1922).
27. Garten, E. Legons sur les Invariants Integraux (Paris, Hermann, 1922).
2B. Schouten, J. A. Rlccl Calculus (Berlin, Sprinoer-Verlaq, 1956).
29. Boullgand, G. Lemons de Gtarttrle vectorleile (Pari*, Vulbert, 1924).
30. Hopf, H. Zum Cllfford-Klelnschen Raumproblem (Hath. Ann.. 95. 1926,
p. 313-339).
31. Levl-Clvlta, T. Absolute Differential Calculus Dover
32. Cartan, E. La Geon6tr1e des espaces de Rlenann (Memorial Sc. math.,
fasc. IX, 1925).
33. E1senhart, L. Pf. Rleaannlan Geometry (Princeton University Press, 1926).
34. Cartan, E. Sur une classe ronarquable d'espaces de Rleaann (Bull. Soc.
math.. 54, 1926, p. 214-264; 55, 1927; p. 114-134).
35. Cartan, E. Sur certalnes formes rlemannlennes remarquables des geometries
a groupe fundamental simple (Ann. Ec. Norm., 44, 1927, p. 345-467).
36. Cartan, E. La thterle des groupes finis et contlnus et 1'Analysis situs
(Memorial Sc. Hath., fasc. XLII, 1930).
37. Cartan, E. Les es paces symetrlques (Verh. Int. Hath. Kongresses, 1, Zurich,
1932, p. 152-161).

INTRODUCTION
by Robert Hermann
Cartan's work has been the Inspiration for much of contemporary
differential geometry, and has Influenced considerable areas of geometric analysis,
mathematical physics and algebraic geometry. This work, which, of course,
began In the 1890's and extended to the 1940's, was In turn based on the great
French and German 19th century geometric tradition, especially Darboux,
Soursat and Lie.
Given these wide ramifications of Cartan's work, any exposIter, general1zer
and "appller" of his work faces the choice of the mathematical setting to
develop the Ideas. In my work, where at all possible, I have chosen the
context of "calculus on finite dimensions, paracompact C" manifolds" treated
with "coordinate-free" methods, since this can be readily understood and
learned (admittedly with considerable effort and fortitude!) by most beginning
mathematics graduate students and those engineers and physicists who have
mathematical background and enough motivation In terns of their own discipline.
Of course, to make serious mathematical progress, It Is often necessary to
Introduce more powerful tools of analysis, algebra or topology, and
understanding the background examples to any specific area often requires a back-tracking
to the more old-fashioned tensor analysis sort of calculations, but I believe
that use of this material Is the best framework for thinking about the widest
variety of areas. (And, In my view, "Cartesian differential geometry" takes
Its richest fom only In the widest possible context.)
Charles Ehresmann was Cartan's student, and was prominent In the efforts
In the period 1935-1960 to develop his Ideas. Motivated partly by Cartan,
partly by Lie and Vesslot, and In part by his own mathematical Intuition
(which was more algebraic and topological than Cartan's), he Introduced
meny geometric superstructures built onto this basic "calculus on manifolds":
Jets and jet bundles, pseudogroups, connections, foliations,... Many of
these Ideas are extremely useful for understanding Cartan, but there Is also
a danger of formalism which outruns the sources of supply In geometric and
physical examples and Intuition. However, anyone doing research or
scholarship In this area must Inevitably use and adopt Ehresmann's Ideas.
One problem has been that his major work of the 1940's and 1950's was published
only In Conptae Fendue notes and In conference proceedings which have long
since disappeared front all but a few libraries. I believe that this problem
will soon be partially rectified by the publication, under the direction of
Hne. Ehresmenn, of his Collected Works. What we still lack Is a high level
exposition In research treatise-graduate text book form of his basic work,
and the related material of his Immediate students of the period, Reeb,
Llbermann, and Haefllger.
373

MOTES ON "GEOMETRIE DES fiSPACES DE RIEMANN*
Carton's work is best understood in terms of what is now called Calculus
on Manifolds, which involves a mixture of elementary parts of point-set and
algebraic topology, real and complex analysis, group theory, and linear-
multilinear algebra. Indeed, this book, along with wayl's Tfut Тала of a
Riemann Surface [1] and Che valley's Theory of Lie Groups [2], is a basic
historical document in its development, we refer also to the mora contemporary
treatises by Helgason [3], Sternberg [4], Hicks [5], Hermann [6], Kobayashi
and Nomizu [7], Kobayashi [8], Bishop and Goldberg [9], Spivak [10], Boothby
[11], Dieudonne [12, Vol. 4], Cheeger and Ebin [13], and Dodson and Poston [14].
The recent bock by Beem and Ehrlich on "Lorentzian" geometry [15] is an
interesting complement to the "Riemannian" ideas.
However, to appreciate the historical background of Cartan's work, it Is
essential to read parts of the great 19th century Cours d'Analyse. Ну favorites
are those by Picard, Goursat, and Valiron. (The latter is not one of the well
known ones, but is on· of th· best for our pruposas. It is tha 1*8fc one, and
the notation and exposition is in a more modern spirit.)
Cartan is, of course, also writing (the First Edition dates from 1926)
in the heyday of tensor analysis. He makes clear, in the Preface to the First
Edition, his opinion that the "debauchs d'indices", which followed the great
popularity of tensor analysis in the 1920's, masked the geometric meaning of
the great ideas of Riemannian geometry· In fact, this book is one of the main
historical documents which inspired the development since 1950 of "global"
Riemannian geometry. (The term "global", used in the 1950's and 1960's
differential geometry, was a code-word to imply that it was not tensor analysis I)
Despite its reputation for wretched excess, tensor analysis does contain much
that ία of both historical and current research interest. My favorites are th·
original article by Ricci and Levi-Civita, translated with explanatory
commentary in [16], and the treatises by Levi-Civita [17], Eiaenhart [18],
Schouten [19], vranceanu [20], and Synge-Schild [21]. (In fact, Vranceanu's
treatise contains extensive material about the relation between the tensor
analysis and Cartan's approach.) The treatises by Misner-Thorne-Wheeler
and Hawking-Ellis contain a stimulating mixture of tensor analysis, modern
geometric ideas, and physics. Tensor analysis still lives in the hearts and
minds of many physicists I What seems to appeal to them is its computational
force, and that, in many problems, one can translate the physical ideas
into "tensor" language with a minimum of formalism. (Of course, "tensors"
ware originally a physical ids*, arising from tha thaory of elasticity.)
375

376
NOTES ON "GEOtfTRIE DES ESPACES DE RIEMANN"
However, this formalism often results in a blurring of the sense of conceptual
clarity which mathematicians prefer. I have found that, having clarified the
conceptual foundations of a geometric (or physical) subject, one can
read the tensor analysis version and pick up шалу useful ideas.
I want to mention to the reader who is not a professional geometer (or
studying to be one) that the search for "generality" in modern geometry
(differential, algebraic, and topological) is not simply a perverse daвire on
the part of mathematicians to make the situation as complicated as possible.
(I admit that it often has this effect I) The image in my own mind is that of
the mountain climber—Don't gat trapped in the underbrush on the lower slopes
of the mountain, but head up the ridges toward the top. Of course, intuition
("geometric" or "physical") is needed to keep one headed in the right direction,
to avoid the false summits. I admit though that often this purity of the
vision of the ultimate geometric goal is corrupted by the necessity of
developing the technical apparatus needed to overcome the crevasses and ice wallsl
For the notation and conventions of manifold theory and related aspects
of fiber bundle theory* Lie group theories, etc., used in these Notes, see
Appendices 1 and 2.
Chapter 1
Sections 1-4
We deal here with a real, finite dimensional vector space E, with a
positive definite, bilinear, symmetric inner product
(х»У) ■* x ■ У .
Suppose Ε is η-dimensional. Adopt the following range of indices, and the
stsmaatlon convention in these indices:
1 <_ i,j,k <_ η .
(However, since we daal here with "Euclidean" geometry, I will use the
summation convention in the following way:
Repeated indices are always summed, whether they are upstairs
or downstairs, unless indicated otherwise. Thus,

NOTES ON "GEOMETRIE DES ESPACES DE RIEKANN" 377
i i 11 A η η
xy - χ у + · ·· + χ у
xiy - Х1У + * * *
Let (e.) be a basis of E. Then, the n»n symmetric matrix <9,.)
with
*ij - ei'ej (1Л)
determineв the inner product terms of this basis.
If χ is an element of E, it can be written in the form:
χ - xiei (1.2)
with χ € R. The χ depend on x, and define real valued functions:
X ■♦ R.
xi(x) - <ei,x> (1.3)
i d
where (e ) is the besis of the dual vector space Ε dual to the е., i.e.
<el,e. > - δ* (1.4)
where
«J - I > (1.5)
(E is the vector space of linear maps: Ε ■* R. The value that an element
β € Ε takes on an e € Ε is denoted as <e ,e> ·) They are a baeie for
the dual space of X, and determine the manifold structure for E.
One can also define the functions x. on Ε as follows:
χ
x, (x) ■ χ · e, . (1.6)
Let (g, .) denote the inverse matrix to (g ), i.e.,
9ijgjk " 6i * (b7)
Then, for χ € E,
χ · et - xj ej ■ et
- xjg
i.e..

378 NOTES OH "GCOMETRIE DCS ESPACES DE RIEHANN"
xi - 9ijxj (1.8)
In tenu of manifold theory, it is most convenient to regard (1.8) as an
identity in the algebra #ЧЕ) of С , real-valued functions on the manifold
E. See Bishop and Goldberg [ 9] for more detail.
Sections 5-12
Here Cartan sketches what became, in the 1930'a and 1940'в, multilinear
algebra. Bishop and Goldberg [91 is the meet useful reference from the point
of view of differential geometry. The part that Cartan uses most is exterior
algebra, i.e., the theory of skev-eyrmetrio tensor*.
From the point of view of this book, where one deals with a finite
dimensional vector space with a "metric", i.e., a positive-definite quadratic
form given, here is one way to proceed.
Let X be the vector space. For χ € X, the inner product ii given.
It can be extended to ХлХ as follows:
хлх is the quotient vector space of the tensor product
Χ β X by linear subspace spanned by vectors of the formt
хОУ + У®*
The image of
— (x®y - y®x)
in this quotient is denoted as
хлу
Remark. Later on in the book, Cartan denotes this ai [x,y]>
The map
(x,y) ■♦ хлу
is then a skew-symmetric, bilinear map
X * X -f Хл X
Extend the inner product to хлх as follows ι
(χχ ax2) . (У1лу2) - i(Xl · yx) (x2 · y2) - Ы1 - у2) (*2 . у )) (1.9)
for х1,х2,у1,у2 € X .

NOTES ON -GEONETRIE DES ESPACES DE RIEMANN" 379
Then, for x,y € X,
|хлу|а - (|х|а|у|Э- (х-у)а) (1-10)
Theorem 1.1.
|хлу| - 2|х||у||в1в θ| (1.11)
where 6 is the real mraber between 0 and 2π such that:
|x-y| - UI|y||«Be| (i.i2)
Proof. Note that formulae (l-ll)-(l-12) are invariant under dilation
у ■♦ Xy
χ + \x ι λ € R , \ * 0 ,
hence it suffices to suppose ι
Ul - |»| - ι .
Then,
4|хлу|2 - (2-2(хву)'(увх))
- |x|2|y|2 ain2 β (1.13)
where θ is the angle between the vectors χ and y.
Suppose (e ) is an orthonormal basis of X, i.e.,
ei ·β - δ (1.14)
i
χ ■ χ β,
i
У - У «i ■
Then,
are a basis far хлх.
Elements s of хлх are called biveotore. Suppose

380 NOTES ON "GEOHETRIE DES ESPACES DE RIEMANM"
with:
Then
• " Ь'Ч, '
t J - - 2J
1 ij
"•Μ " 2Z eij'ekl
■ i"1J (^ν*ι*ν
1 2 I
2 к Д " 2 ztk
The rest of the material in Sections 5-18 may be treated in a similar
algebraic way.
The basic geometry underlying this algebra is the theory of the Graeemann
manifolds of the vector spaces Ei
m
G (E) - eet of m-dimensional linear eubspaces of Ε
The Lie group of linear automorphisms of Ε which preserves the inner
product acts transitively on GlE) and plays a basic role. It is
isomorphic to 0(n,R}« the η χ η real orthogonal matrices. G (E) is the
coset space 0(n,R)/0(m,R) *0(n-m,R).
Section 19
A moving or free vector of Ε may be considered as an element of the
tangent vector bundle to e, T(E). T(E) may be identified with
EXE .
A pair (X/V) £ ΕχΕ is identified with the tangent vector at t - 0
of the curve
t -» χ + tv .
The projection map T(E) ■* Ε is just
(x,v) ·* χ .

NOTES ON "tfOMETRIE DES ESPACES DE RIEHANN" 381
The principal bundle or from bundU to Ε is the set of (n+1)- tuples
(xi v, ,...,v) *
1 η
X» У,,...|Уя € Ε ,
ι η
such that:
ν, л ... л ν ^ 0 .
1 П
The "p-vecteurs glissants" form an associated bundle ι An element of this
bundle is a pair
Section 20
This section injects a brief dose of physical intuition. This is quite
coHon in Cartan's expository work. Often, J have found these "bits" to be
of enormous significance. For example, in the book Ьваопе виг Ιββ Invariant*
Integraux one can find such of the present dey "symplectic structure" approach
to analytical mechanics laid out in Cartan's remarks.
In this section Cartan alludes to the geometric nature of "angular
city
product
velocity" of a rigid body moving in R : It is an element of the exterior
T(R3) л T(R3)
of the tangent bundle of the manifold of "our" three-dimensional Eulidean
space. Йоге concretely, it is a pair
(χι ω)
consisting of a point χ of R (the "instantaneous fixed point" of the
rigid body) and an element
ω € R3 л R3
X X
of the exterior product of the tangent space of R at χ with itself.
Since R is a vector space, its tangent space can be identified with itself.

382
NOTES ON "GEOHETRIE DES ESPACES DE RIEHANN"
Further, each element of the exterior product of two three-dimensional vector
spaces is eimple, i.e., can be written as
ω - β л β
β1'β2 € Rx = R
These elements are called biveotore.
Sections 21-26
This material is now covered in tmiBor algebra. The treatise by Bishop
and Goldberg (9] is a standard reference for this material.
Chapter II
Sections 27-31
Let Ε be a vector space with a positive definite quadratic form
(x,y) -* χ · у (2.1)
Ε «Ε ■+ R .
Since the tangent bundle T(E) is identified with a product)
T(E) - Ex Ε
- {(X.V): x€E, v€E)
i.e., Ε is parallulizabl*, the inner product (1.1) defines an inner product

NOTES ON "GEOttfTRIE OES ЁSPACES DE RIEMANN"
333
on the tangent bundle,
(χ,ν Ϊ · (x,v2» - ν · ν . (2.2)
A geometric way of thinking of it is that the tangent space at each point
χ € Ε is translated to the origin О € Ε, where it is identified with Ε
itself/ and then the inner product is * is carried over.
Analytically* this construction can be described as follows. Let
(ei) . l<i,j<n
be a basis for the vector space E. Set;
Let (x ) be the functions such that
χ - χ (x)e,
for χ ε ε .
i »
The (x ) are real-valued С functions over the manifold E:
χ ·* (x (x),. ..,x (x))
is a diffeomorphlsm Ε - R , The (x ) form a coordinate system. Then
dx1 € ^(E)
are one-differential forms on Ε (cross-sections of the cotangent bundle),
which
Then,
which form a basis of 0 (e) , considered as a module over ДОЕ).
χ · χ - g., χ (x)x3(x)
Set:
de2 - g dxidxJ (2.3)
where the product between differential forms on the right hand side of (2.3)
is the symmetric product
(dxidxj)(v1,v2) - j (xi(v1)xj(v2) +xi(v2)vj(v1) (2.4)
for χ € ε, ν1#ν € Εχ ,
2
Then, (2.4) defines "de " u a field of positive definite inner products over
the tangent vectors to X. This is a Riemccnnian metric.

384
NOTES OH "GEOMETRIE DES ΕSPACES DE RIEHANH"
New, one can use other coordinate aye tests for E. Choose
i
u
as elements of ДОЕ) (or of £"(D), where D is an open subaet of E) such
that ι
du л ... л du ^ 0
Then, the dx can be written as coctoinations of the du , and this can be
substituted into (2.3) to write;
ds2 - g* du*du^ (2.5)
where (g',) is another symnetric matrix of functions. The question Cartan
asked in this chapter is:
Given a Riemannian metric in the forn (2,4), how can
it be recognized as coming from a quadratic form?
Sections 32-33
то answer this question, cartan introduces the method of the moving from*
Given the inner product · , let
E(X) - {(x; v, v) Cx"*1, v,a...av * 0 ) (2.6)
In in
(X denotes the Cartesian product of n + 1 copies of X.) E(X) is the
bundle of orthonormal frame.
Let G be the Lie group of η * η real invariable matrices. (Denoted
as GL(n,R) in the Lie group literature.) G acts on E(X) as follow*: If
9 * (g^K
gUi v,,...,v Ϊ - (xi g,v.,.. -,g v.) (2.7)
1 П 1 i П i
Let
N: E(X) -*■ X
be the nap such that:
M(xi v.,...,v ) ■ x
ι η
Μ is a Bubrwreion map (i.e., the linear map N indiced on the tangent bundle
is onto), and the fibers are the orbite of G. G acts simply and effectively
on the fibers: Mi E(X) -*■ X defines E(X) as a principal fiber bundle with
G as structure group.

NOTES OH "GEOMETRIE DES ΕSPACES DE RIEMANN"
385
We can also consider N as a vector-valued function on E(X) , i.e., as
lament of 4
■spai E(X) -*- X:
an element of ДОЕ(Х))0Х. (See Appendix 1.) Define β,ι...ιβ as similar
β.((χϊ (ν,ι...,ν )) - v. ,
1 1 П 1
i - Ι,.,.,η .
We can now exterior differentiate N and e as elements of JF(E(X))0X.
dM - u β
de - «.β-*
(2.8)
The u ,ω are one-forms on E(X).
Mow, suppose that D is another m-dimensional manifold with coordinates
(uS .
Let
φι D -*■ X
be a diffsomorphism. Define a map
OIU + E(X)
as follows ι
a(u) - (ф(и), e1(u),...,en(u)) (2.9)
where
d**(M) - duiei
i.e.,
V- - fr ■
3u
Then, the pull-back of relations (2.8) (always
considered as an X-valued differential forn) isi
d**(M) - due.
d**(ei) - ♦*(»{)ej .
(2.10)
These are the relations Cartan considers in Section 32.

386
NOTES ON "GtotfTRIE DES ESPACES DE RIEMANN-
Section 33
Now, impose an inner product on X. Set:
g.. (u) - e (a(u)J · e (a(u)J
- α*(βι·β)
The Riemannian metric
g. . du du
iD
2
is then the pull-back via φ of the Riemannian metric ds that the inner
product · defines on X:
gij du*duj - φ*(da2)
Now, the Chriatoffel eymbole are the components of the scalar-valued
one-forms
and
wij " 9ikwj
on 0 with respect to the basis du of one-forms
α*(ω^ϊ - Γ* du*
-ij " Pijkduk '
in Section 33 Cartan derives the classical expressions for the Christoffel
symbols in terms of the "metric tensor" g ,.
Sections 34-42
Cartan continues to develop what were, in 1926, the standard formulas of
tensor-analysis Riemannian geometry for the case of a Riemannian metric
on U resulting from pulling back the metric on the vector space X
associated with a positive definite inner product *, by means of a
diffeomorphism φι U ■+ v. in the classical literature, this is often what
is meant by using curvilinear coordinates.
Sections 43-49
Now, Carten gears up and asks for the converse* i.e., when does a
Riemannian metric

NOTES OH "GEONETRIE DES ^SPACES DE RIEMANN
387
as ■ g , du du
arise fro» a diffeoaorphism φι U ■+ X. Recall thati
do*(u) - du α*(β.ϊ
dei - "i \
α*(ωιϊ " rijwJ
(2.11)
(2.12)
(2.13)
Exterior differentiating (2.11) gives the structure equations for the affine
group:
i ω, л ω
with the map
Oi U ■* ΕΐΧ)
given as ι
a(u)
■(■
♦ (u).
*L·
li-
f)
Эй Эй
Combining relations (2.11)-(2.14) gives Cartan's Equation (28):
эг
Эй
ЭГ
Эй
it + Гп Гк _ Γη Гк - о
(2.14)
(2.15)
Now, he asks for the converse, i.e., do relations (2.15) for a ds
given a priori guarantee the existence of the maps φ and a. He proves
2
this (if the g,, are of class С , and if U is simply connected) by
defining the manifold
Ζ - UxE(X)
and writing down following X-valued one-forms on Z:
dM - due,
dei - r« du4
(2.16)
He has now set things up so that conditions (2.15) are the conditions
for complete integrability, in the Frobeniue sense, for the Pfaffian system
which the forms (2.16) define. This defines a foliation of DxE(X): the
leaves are of the same dimension as U, and project down in a local

388
NOTES ON "6E0HETRIE DES £SPACES DE RIEVMNN"
diffeomorphic way to U. Thus, if U is «mall enough, the maps φι U -► X,
a: U ■* E(X) will exist, as the inverse maps to the mapi
Leaf of (2.16) ■♦ U
In the next chapter, Cartan investigates in more detail and depth the global
sufficient conditions for the existence of φ,α as a consequence of the
"flatness" conditions (2.15).
Chapter III
Sections 50-52
Definition of Rimmamxan mini fold, in more-or-less modern sense.
Sections 53-56
* 2
Let X be a (C ι pexacompact) manifold with a Riemannian metric ds .
X is XooaXXy Euotidacot if, in each local coordinate system, the identities
(2.15) are satisfied. This amounts to saying that for each χ € X, there is
an open subset U of R and a diffeomorphism
φι U ■* Χ
χ ε Ф(ш ,
such that:
φ* (da2) - g duiduj
alone (g ,) is a constant, positive definite, symmetric matrix. "Normal''
means "complete", in the modern sense, i.e., that one of the conditions of the
Hopf-Rinow [6] theorem be satisfied.
Sections 57-78
Here is what Cartan proves in modern language. Let X be a complete,
locally Euclidean Riemannian manifold. Let X' be its simply connected
covering space. Lift up the metric to X'. It is again complete. X', with
this metric, is then globally isometric to R , with the Euclidean metric on
R . As a consequence, X can be written as an orbit space
X - H\RD ,
where H is a discrete group of i some tries of К , which acts effectively.

NOTES ON "GEOMETRIE DES ЁSPACES DE RIEMANN"
389
The torus and cylinder for η - 2 are the obvious examples. See the treatise
by J. Wolf [22] for the standard, modern account of this material.
Chapter IV
Sections 79-92
In these sections, Cartan begins the theory of general Riemannian geometry
along classical lines. Let X be a manifold with a given Riemannian metric
2
da . Consider a point χ € X, an open set U containing χ , and a diffeo-
norphism:
♦: U - Rn .
Let ds be the standard metric on R . Consider the quadratic differential
form:
ds2 - T*(ds2) . (4.1)
One can use φ to make geometric constructions and definitions. φ is said
to be OBOulattng if the form (4.1) and its first derivatives (i.e., its one-
jet) vanish at xQ. In these sections, Cartan points out that the
construction of Euclidean differential geometry which were reviewed in the
previous chapter and which only depend on the metric tensor g.. and the
Christ of fel symbols are independent of the ohoiae of osculating metric.
Sections 92-98
Bore Cartan develop* an idea of the physicist Enrico Fermi. (It is
interesting to see this concrete illustration of how, in the 1920*s, Einstein's
work stimulated intellectual collaboration between physicists and mathematicians!)
2
Given the metric ds and a smooth curve σ: [0,a) ■+ U such that σ(0) - χ ,
one can choose φ such that the one-jet of difference quadratic differential
form (4.1) vanishes at every point of a. Again, this enables one to study
certain geometric properties of curves in general Riemannian manifolds by
reference to properties of curves in R with the standard metric.
Sections 99-100
This is the beginning of the development of the Levi-Civita affine
connection associated with the metric. See [6] for my version of this.

390
NOTES OH NG£0HETRIE DES £SPACES DE RIDWflT
Chapter V
Sections 103-106
Let X be я manifold with a given Rienannian metric. In Section 86
Cartan has defined geodesic curves a* the solution of certain second order
differential equations. Through each tangent vector there is a unique geodesic.
A submanifold Υ is said to be geodesic at the point xQ if each geodesic
of the metric on X beginning at χ and tangent there to Υ lies, at least
for small values of the parameter, in Υ. Υ is said to be totally geodesic
if it is geodesic at each of ite points.
Here is another way of stating this. A Rienannian metric determines a
torsion-free affine connection, the Levi-Civita connect ion. One can use this
connection to define when a vector field
t - v(t) € Xo(t)
along e curve
t -. o(t)
in X is self-parallel. (Namely, the со variant derivative of the vector field
t -* v t(tlv(t), ehould be aero.) a curve t ■* o(t) is geodesic ifi
V0.(t) a'Ct) - 0 . (5.1)
2
Let φ: Υ ■+ X be a submanifold map. The metric ds on X can be
pulled back to define a metric
♦*(ds2) (5.2)
on Y. The metric (5.2) determines an affine connection on Y, benoe a family
of geodesies. φ is said to define Υ as a totally geodesic eubnunifold of
X if it maps every geodesic of the metric (5.2) into a geodesic of the
2
metric ds .
Section 106
The theorem of Ricci that Cartan cites is related to a theoree of mine l23l
about what one now calls "Rienannian foliations with totally geodesic fibers".
Sections 108-116
These are classical results about geometric properties of Rienannian
manifolds of constant curvature.

NOTES ON "GEOMETRIE DES ίSPACES DE RIEHANN"
391
Chapter VI
Continuation of classical theory of spaces of constant curvature. This
is done very beautifully, and I have nothing to add.
Chapter VII and VIII
Sections 158-163
Let U be a coordinate neighborhood of the Riemannian manifold X,
with coordinates (u ) a coordinate system for U, with
2 i j
ds « g.. du du
the metric tensor. Let
JK1 * \&i au au /
be the Chriatoffel aymbola of the first kind.
Set ι
Uij - rijRduk (7.2)
Wi " gkJwji ' (7'3)
Cartan is still working with the unnatural (from the point of view of
Riemannian geometry) "affine" frames, so he must continue to use upstairs and
downstairs indices, together with the usual tensor analysis summation convention
Let
E(Rn) - {(Mj β,(...,β ): М,,в,(...,в €r", ·,Α...ΑΒ ψ- 0 )
1 η 1 1 η 1 η (? 4^
Definition. Let t ■+ o(t) be a curve in U. A curve
t -. (M(t), e.(t),...,e (tj)
ι η
in E(R ) is the development of α if the following differential equation
is satisfied)
*M d . k, lt_xx
- - -(u (o(t))ek
dt— ■ MdiK ■
(7.5)

392
MOTES ON "GEOMETRIE DES ESPACES DE RIENWW
where t ■+ do/dt is the tangent vector to the curve.
As we have done before, let us introduce the following R -valued one-form
in UxE(Rn):
θ - dM - duke
(7.6)
9i " *i * Vk '
Then, Car tan essentially deals with curves in U*E(R ) which are Integral
curves of the Pfaffian system generated by θ,θ , i.e., their tangent vectors
annihilate θ and θ.. Cartan considers the integral curves which project
into a given curve σ in U, and calls their projections into E(R ) the
development of σ.
Now, suppose we have a honotopy on U, i.e., a mapping
(t,s) - o(t,s)
2
of R into U. Let
Э o(s,t) - tangent vector to the curve u + o(u,s)
(7.7)
at u - t
Э o(s,t) m tangent vector to the curve u ■+ o(t,u)
at u - s.
In my treatise [6], which was, of course, strongly influenced by my
reading of Cartan, I based my treatment of the analytical aspects of Riemannian
geometry on this "partial derivative" formalism. In the late 1950's,
Warren Ambrose was instrumental in developing the modern aspects of Cartan's
formalism on Riemannian geometry, and I learned much of the material that
appeared in [6] from him.
Construct the corresponding homotopies in E(R ), as solutions of the
following differential equations:
3t " It (U (0(e'tn \
(7.8)
tai к
ЭМ Э , k. fcll
- - -<u (s,t)) eR
(7.9)
Эв1 к
— - UiOso(t,s)) .

NOTES OH "GEOtfTRIE DES ίSPACES DE RIEMANNH
393
Cartan wants to compare solutions of (7.8)-(7.9). This Involves the
exterior derivatives of the one-forme (7.6), which we will now calculate ι
do * - du a de.
к ч
■ - du α (Θ. + uj'e.)
■ - du Α Θ. - du α ωΓβ
(7.101
к к
'Л * "Ι
dbi.e, + ω, л de.
- -*ί% + "ϊΛ (ek+ukej)
" ω1 A 9k * °iek
with
°i " ' **i * "i A ^ (7,11ϊ
Riil *** A dul (7.12)
the ourvature forme, R the Riemann-Chrietoffel ourvature tensor.
Sections 164-199
Again, having developed a formalism half way between classical tensor
analysis and modern manifold theory, Cartan develops traditional material in
an optimally clear and beautiful form. It is only In the next chapter that
Cartan began to make use of essentially new methods, the famous Method of the
Moving Frame.
Chapter VIX
ЯAction* 201-204
The Riemannian geometric aspects of the method of the moving frame
essentially involve two sorts of frame bundles: manifolds that we will call
F and F (the Riemannian or orthonormal frame bundle) and the affine
frame bundle, F has a basis of one-forms (i.e., an absolute parallelism)
labelled
(iAmJ) , 1 <. i,j < η (9.1)

394
NOTES ON "tfOMETRIE DES ESPACES DE RIEHANN"
The following structure equations are postulated!
d» - ω Α ω3 (9.2)
duj - ω* Α ω|| + Q^ (9.3)
Ql - ϊ "Ϊιμ-k д "* (9-4>
2
F has dimension η + η . The Pfafflan system
ω1 - 0 (9.5)
2
is ootrpletely integrabl*. Its leaves are of distension η . Restricted to
each leaf, the foroa vr fore a basis Θ. of one-forme which satisfy the
structure relations of GL(n,R) ι
d9i " 9i A 9k (9,6Ϊ
Let \r be the vector fields on F. which satisfy the following relations ι
i A
ω1^) - 0 (9.7)
■}iV*> - β# (9.8)
The (VT) then fore a finite dimensional Lie algebra of vector fields
isomorphic to the Lie algebra of GL(nrR). If the system is complete (e.g.,
in the sense that the Riemannian metric
i i i i
ω ω + ω.ω,
is complete), then the v~ generate a Lie group G which acts transitively
and almost effectively on the leaves of the foliation (9.5). Let
X - С\РД (9.9)
be the orbit space. If the foliation (9.5) is regular, X is a paracompact,
Hausdorff manifold. F defines a certain geometric structure on X,
called an affine connection. One can work locally on X by choosing open
sets U, and cross-section naps
Ϊ! U ■* F . (9.10)
γ is called a moving frame. In this book, Caxtan defines a natural moving
front as one which satisfies ι

NOTES ON "GEOMETRIE DES ESPACES DE RIEMANN"
395
γ'ίω1) - du1 , (9.11)
where (u ) is a coordinate system for U.
F is a manifold of dimension
n(n-l) n(n+l)
η + -
2 2
with an absolute parallelism of one-forme labelled
(ω^ω^ϊ (9.12)
ωϋ * U1i " ° (9.13)
such thatt
du - ω Α ω (9.14)
""il " "ii A "kj + вц (9Л5)
aH " 7"им"кА"1 (9·16'
Again, the structure equations (9.14)-(9.16) imply that the Pfaffian system
ω - 0 (9.17)
is completely integrable.
Label as Θ.. the form ω.. restricted to a leaf of the foliations.
They satisfy the structure equations of the orthogonal group 0(n,R) - G.
Let V.. be the vector fields such that:
w (V ) - 0
il jk'
"ij'V " β1Χβ11
(9.19)
The (V..) then generate a Lie algebra of vector fields isomorphic ίο the
Lie algabra 0(n,R). If the metric
ui ' ui * uij ' uij
is complete and the foliation (9.17) is regular» G acts on F and the
orbit space
X - G\FQ
(9.20)

396
NOTES OH "GEOMETRIE DES Ε SPACES DE RIEMWM"
is a nanifold. U is an open subset of X, with
v. u - p0
a cross-section, then the Riemannian metric
ds2 - γ*^) -γ·^) (9.21)
2
is intrinsically associated with the structures, i.e., ds is independent of
the cross-section map γ.
There are, of course, interrelations between X, F , and Γ , and
conditions which determine one uniquely in terns of the other or the metric
(9.21). I will defer to a later publication for a full development of this
sketch of a theory. (See [25-27] for relatad material.)
Xn greater generality, a space x with a Carbon oann*otion otth Lie
structure group G involves the following datat
a) A manifold F.
b) An action of G on Г which is effective.
c) The orbit space
X - G\F
is a manifold.
i a
d) P hoe an absolute parallelism of one-form (и ,u ),
1 <. i»J <. η ■ dim X, 1 <^ a,b <^ m - dim G such that
ω - 0 is completely integrablej the orbits of G
are the leaves. The vector fields V such that:
ω1 (V) - 0
(9.22)
d(taia(V)) - 0
generate the Lie algebra of G.
e) There are constants λ and functions Τ such thats
dw1 - a* uj л и* + \\ «j A ui* + T* Hj A wk (9.23)
jk ja jk
du* - A.. ω"1 A «k + λ* J* А ыЬ + λ* wb A «°
jk jb bo (9#24)
a j к
+ R.1. май
jk

NOTES ON "GEOMETRIE DES ίSPACES DE RIEMANN
397
However, this book does not go into thee· more general sorts of spaces.
Part III of Cartan's Collmatmd Vorka deals with them.
С Ehresmann, in his classic work [24], formalised this Cartan connection
material in a certain form. (See also [7,8].) However» there is a difference
in methodology, if not substance, between Cartan and thee· later workers.
They have tried to be determinedly index-free. Personally, I like indices
when they axe used judiciously (and Cartan provides a model for thisl). There
is another admirable feature in Cartan's approach: It is designed to do
certain types of computations in what I believe is almost optimally efficient
form.
Sections 205-211
Return to the case where:
G - O(n.R)
Τ - 0
0
as described above, with a basis of forms labelled
(νν
(9.25)
"ij + "ji " °
satisfying the Riemannian structure equations.
What Cartan calls the Method of the Moving Frame involves studying sub-
manifolds of X, the quotient space, by setting up certain Pfaffian (or,
more generally, exterior differential systems) on F involving constant
coefficient combinations of the forms (9.25) and looking for submanifolds of
F which annihilate these forms, and project into the required submanifolds
of X. See [4] for a start at this material in more modern terms.
Chapter X
Sections 213-219
Here is a modern version of some of this material. Let X be a manifold
with a given Riemannian metric. Assume, for simplicity, that it is complete,
i.e., geodesies can be indefinitely extended. (Cartan calls this normal.) Let xQ
be a point of X.

Мь
NOTES ON "GEOHETRIE DES ЁSPACES DE RIEHMtfT
Let Χχ^ be the tangent space to X at χ . Let
Exp: X + X
χ
0
be the exponential map. It is defined as follows:
For ν С X , Exp (ν) is the end point of the geodesic
which began at γ , has ν as tangent vector* and has
length equal to the length (in the quadratic form defined by
the metric) of v.
Another way of putting it is as follows:
For each ν € Χχ.»
t -*· exp(tv) , 0 <_ t < - ,
is the geodesic curve which began at χ and is
tangent there to v.
One proves that the nap exp is a local diffeooorphism at the sero point
of Xjj.- Thus, there is a nieghborhood of О on the vector space which is
mapped diffecnaorphically under exp to a neighborhood of χ . One c«n take
this neighborhood of 0 to be the set of all vectors ν С Χχ»» such that
< r
where r is a positive real number. This open set of X is called a
ввго ball of radius r. It* basic geometric property is that each point
of it can be joined to χ by a unique geodesic of length less than r.
Let (v.) be a basis of 7L. such that
ι 4)
V, * VJ - 0, .
ι J ID
i.e., (v ) is an orthonormal form with respect to the inner product on tangent
vectors. Let 8 (x ) be a geodesic ball of radius r. There is then a basis
of vector fields (V.) on 8 (x ) such that
w ■ vi
For each ν € X- , the vector fields V. are
self-parallel (i.e., of invariant derivative sero)
along the geodesic curve t -*■ exp(tv).
Let ω be the dual basis of one-forms in 8 (χ Ϊ.

NOTES ON "GEOHETRIE DES ESPACES DE RtEHANN"
399
The metric is then;
л 2
da - ω · ω .
Let ω . be the associated connection form in Br(xQb i.e., the one-form
such that
*"i - "uA"i
uij + uji ■ ° ·
For each t between 0 and 1, let
ExpS X + X
Ж0
be the map defined as follows:
Sett
Exp (v) *- Exp(tv) .
-t t*
ω - Ext (ω Ϊ
(10.1)
-t t, ,
ω - Εχρ (ω i
The forms (u.,u..) are then defined in a neighborhood of zero on the vector
space Χ- . Equation (7) of Section 217 determines these forms by means of
linear ordinary differential equations in t, in terms of the аощопепЬе
curvature tensor in this frame.
These equations are a basic tool in Cartsn*s work, particularly in his
theory of eyrrmtric spaces, where Equations (7) became constant coefficient
linear differential equations. They show how the metric and the Levi-
Civits connection is determined by the curvature tensor.
Section 220
Here* Cartan makes the equation for ω. explicitly in case the curvature
is constant» thus obtaining classical formulas for the metric tensor of spaces
of constant curvature.

400
NOTES ON "GfOHETRIE DES CSPACES DE RIEMANN"
Section» 223-233
Here Cartan derives approximate solutions of the differential equations
for the forma u, > ш**· *nd uses these to praaent classical material on
Riemannian geometry. This is the beginning of the Rauch Comparison
Theorem 113]/ which compares geometry and topology of Riemannian manifolds
of variable and constant curvature.
This Rauch material is one of the highpoints of post World War II (and
post Cartan) geometry/ and to a certain extent* served as the mathematical
inspiration for the Hawking-Penrose work on singularities of Lortenr manifolds
115].
Chapter XI
This chapter briefly presents what is now recognised as one of Cartan'β
greatest works * thm theory of eymmtria враола. Refer to the treatises by
Halgason I 3] and Kobayashl-Nomisu 17 ] for more detail.
Again/ this material has become one of the highlights of 20th century
geometry/ having surprising and marvelous ramifications in harmonic analysis
(e.g., work of Ηarish-Chandra Galfand, et al.), number theory, algebraic
geometry, and so on.
Chapters XII, XIII, Notes 1-5
The rest of the book, is really an account of Cartan's research on
Riemannian geometry and Lie group theory. I will only sketch a general
setting here for some of Cartan's ideas on the relation between Cartan
geometries (Riemannian/ affine, conformal, projective, etc.) and groups of
symmetries.
Let Ζ be a manifold of dimension η + m, with an absolute parallelism
ω , 1 <^ u,v <_ η + ш (12.1)
of one-forms. Suppose X is a manifold of dimension n, and let
w: Ζ ■* X (12.2)
be a submersion mapping, whose fibers лх· the leaves of the foliation
ω -Ο, Ι^α^η. (12.3)
Suppose
* Чу " (12.4)

NOTES ON "GEOMETRIE DES ^SPACES DE RIEHANN"
401
with
df? - £ ϋΛ (12.5)
βΎ Βγ.Υΐ
βΎ'ΪΙ βΎ*Ύΐϊ2
For each ζ Ε Ζ, let
Λ(ζ) - set of points ζ' С Ζ such that
βγ*ϊΐ βγ*ϊι
ΒΥ·Υ1Υ2 βϊ»Ύΐ/Ύ2
Let G be the group of diffeomorphisms gs Ζ -*■ Ζ such that]
g*(Ui) - ω1 (12.в)
1 1 А*Э i n + я
Thus* G is the group of automorphisms of the absolute parallelism. G also
preserves the foliation (12.2). Hence, G maps each fiber of * into another
fiber» and acts on X.
(g»x) -*■ gx .
The map ν then intertwines these actions of G on Ζ and x.
Now, G acts effectively on Z, i.e., if g € G leaves one point of Ζ
fixed, g is the identity. The following properties can be proved:
G is a Lie group. (12.9)
The map
GxZ-»Z
(12.10)
(g#z) ■* g*
is proper.
G(A(Z)) С A(g) (12.11)
for each s € Ζ

402
NOTES ON "tfOHETRIE DES ^SPACES DE RIEMANN*
Under certain hypotheses 127] one can prove that a
Gz - Λ(ζ) (12.12)
for each ζ € Ζ ,
i.e., the orbits of G on Ζ are precisely the subset*
obtained by setting equal to saro all the structure functions
rt and their "covariant derivatives" f? #...
&У βϊ.ϊι
Many of the special calculations of Cartan in this last part of the book
are designed to calculate g or G in this way for various Z's associated
with Riemannlan manifolds X.
The simplest special case is that where:
X - a Riemannian manifold of constant Riemann curvature
ζ - orthonormal frame bundle of X.
The absolute parallelism of Ζ is that defined by the Levi-Civita affine
connection, labelled:
(ω.,ω.,) ,
3 (12.13)
ω. . + ω.. ■ О
Ζ is of dimension
A n(n- 1) (n+ l)n
η ♦ - - -2
The structure equations (which taXe into account that the metric on X
is of constant curvature K) are:
A,\ ш и Л 111
i ij j
du.. - ω Α ω. . + Κω. Α ω. (13.14)
Thue, the subsets A(z) are all Ζ itself, i.e.* the structure functions
are constants. Equation (12.14) then expresses the fact that the (ω,,ω, Ϊ
satisfy the Cartan-Maurer equations for SO(n+l,R), which is the group of
isome tries of a Riemannian manifold of constant (positive) curvature.
Another special case is:
η - 2 (12.15)
(12.14) is then the structure equation of a two-dimensional Riemannian
manifold with Gaussian curvature K. (In this dimension, the Riemannian

NOTES ON "GftJHETRIE DES I SPACES DE RIEHANN1
403
curvature tensor пая only one component, which is just K, and which «ay be
a variable function. In the case η ^ 3, relations of the font (12.14) plus
the Bianchi identities, imply К - constant.)
In dimensions η - 2 and 3* as Cartan shows* these general facts can
be used to analyse the possible structure of G and the font of the curvature
tensor in the variable curvature case. Such results (particularly when
extended to peeudo-Riemannian situations) have been useful to physicists in
General Relativity.
Of course, a special feature of Cartan's work is the use of these frame
bundles to study the geometry of manifolds. Unfortunately* this is (in my
opinion) the hardest part of Cartan's work to understand. In part, this is
an "intrinsic" difficulty. Cartan was deeply familiar with the 19th century
geometry of curves and surfaces (exemplified in Darboux' Thsorie dee Surfaces)
and much of his work here represents a working out of the 19th century material
in his framework. Thus, to understand it completely also requires a scholarly
knowledge of this 19th century literature which, alas* is now lost.
S.S. Chern's work 128] has been the most extensive and notable in carrying
on Cartan's legacy on the geometry of submanifolds. H. Guggenhelmer's book
Differential Geometry 129] is the most accessible exposition of some of this
material for the modern reader.
References
1. II. Weyl, The Concept of a Riemann Surface, Addison Wesley, 1965.
2. С Chevalley, Lie Groups, Princeton University Press, 1946.
3. S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press,
1961.
4. S. Sternberg* Leatures on Differential Geometry, Prentice-Hall* Englewood
Cliffs, N.J. * 1964.
5. N. Hicks, Notes on Differential Geometry, van Host rand, 1965.
6. R. Hermann, Differential Geometry and the Calculus of Variations, 2nd Ed.,
Interdisciplinary Mathematics, Vol. 17, Math Sci press* Brookline, MA*
1977.
7. s. Kobayashi and к. ttomisu* Foundation of Differential Geometry, Vols, ζ
and II, Interscience, New York, 1963.
8. s. Kobayashi, Transformation Groups in Differential Geometry, springer-
Verlag* Berlin, 1972.
9. r. Bishop and s. Goldberg, Tensor Analysis on Manifolds, Dover
10. M. Spivak* Calculus on Manifolds, W.A. Benjamin, New York, 1965.
11. N. Boothby, Differential Geometry, Academic Press* 1973.

404 NOTES ON "GEОМЕTRIE DES ΕSPACES DE RIEHANN"
12. J. Dieudonne, Treatise on Analysis, Academic Press, 1973.
13. J. Cheager and D. Ebin, Comparison Theorem in Riemamian Geometry»
North-Holland, 1975.
14. С Dodson and т. poston. Tensor Geometry, Pitman, 1977.
15. J. Been and D. Ehrlien Global Lorenteian Geometry, H. Dekker, New York,
1981.
16. R. Hermann, Ricai and LevC-Civita'в Tensor Analysis Paper, Lie Groups*
Vol. 2, Math Sci press, Вrookline, MA, 1975.
17. T. Levi-Civita, The Absolute Differential Calculus, Blackie, London,
1928 (Dover Publications, New York) .
18. L. Eisenhart, Riemannian Geometry, Princeton Univ. press,
19. J. schouten. The Rieai-Calaulus, Springar-Verlag, 1955.
20. G. Vranceanu, Leoons de GUomAtrie DiffUrentielle, Academic da la
Republique Populaire Roumaine, 1957.
21. J.L. Synge and A. Schild, Tensor Calculus, Dover Publications, 1978.
22. j. Wolf, Spaces of Constant Curvature, McGraw-Hill, 1960 (now published
by Publish or Perish).
23. R. Hermann, A sufficient condition thet a map of Riemannian manifolds
be a fiber bundle. Pre». Am. Math. Soa. lb 236 (1960).
24. С Ehresmann, Les connexions infinitesiaales dans un ίβρβσβ fibre1»
Colloque de Topologie, Bruxelles, 1950.
25. R. Hermann, Cartan connections and the «quivalance problems for geometric
structures. Contributions to Differential Equations 3, 199-248 (1964).
26. R. Hermann, Equivalence of submanifolds of homogeneous spaces. Math. Ann.
158, 284-289 (1965).
27. ft. Hermann, Existence in the Large of parallelism hoaomorphisms, Trans.
Am* Math. Soa. 161, 170-183 (1963).
28. S.S. Cnern, Selected Papers, Springar-Verlag, 1978.
29. H. Guggenheimer, Differential Geometry, Dover Publications, New York,
1977.

APPENDIX 1
THE FORMALISM OF CONNECTION THEORY
1. PFAFFIAN AND EXTERIOR SYSTEMS
All manifolds, шаре* and geometric data will be С , finite dimensional
and paraoonpact, unless mentioned otherwise. Let X be such a manifold. Here
is the notation we shall use for the basic objects of the calculus of manifolda
used in this work.
ДОХ) - algebra of С * real-valued functions on X.
For χ С X/ X - tangent vector space to X at ж.
d
X - dual space to X ,
X X
- space of one-covectors at x.
T(X) - { (x,v) ι xEX, VEX )
■ tangent vector bundle
Td(x) - { (x,e): x€x, eexd)
- cotangent bundle .
У(Х) - С* cross-sections of T(X)
- derivations of ДОХ) .
0 (x) ■ cross-sections of τ (χ)
- dual ДОХ)-module to V(X)
0 (X) · exterior differential forms of degree η
d: βη(Χ) -*■ Φη (x), exterior derivative
(Vl'V "* IV1'V2] ' V1V2 * V2V1' *" R"1111·" "*Ρ
I . ]i ΨΊΧ) « У(Х) ■* fix) , called Jaoobi bracket.
405

406
APPENDIX 1 — CONNECTION THEORY
If θ is a differential for» and V is a vector field, then
#ν(θϊ
denotes the Lie derivative of θ by V.
G (X ) - Grassaann manifold of m-dlmensional linear subspaces
of the vector space X , m - 0,1,2,...
G*(T(X)) - the fiber bundle over X whose fiber over χ € X
is G*(X ϊ .
G (Τ (Χ)) - bundle of m-dimensianal linear sub spaces of the
cotangent vector spaces .
If Ε ■* X is a fiber apace over X, let Γ(Ε) denote the space of С
cross-section maps: X -*■ E.
Definition. A Pfaffian eye tern (of diaension m) on X is a cross-section map
Υ: Χ ■* G*(T(X)) ,
i.e., an element of Γ(σ"(Τ(Χ)).
Given such a cross-section γ € Γ(θ™(Τ(Χ))), one can define the "dual"
object.
Yd €r(Gn-a(Td(x)))
γ (X) - space of θ С X such that θ(γ(χ)) - 0
- annihilator of γ(χ) in the dual space
d
γ is actually the object which Cartan would call a "Pfaffian system", since
he preferred to work with differential forms rather than vector fields.
However, we will call either γ or γ a "Pfaffian system". (When one
considers such systems with singularities» it is necessary to distinguish one
form its dual.)
Set:
ΤΚ(γ) - the set of vector fields v€ fix) such that
V(x) С (χ), for all χ С X .
Thus, the orbit curves of the pfaffian system are the curves in X that are
orbit curves of вате vector field on ΊΗγ). (Nots that ΊΗγϊ is not a Lie
subalgebra of У(Х), unless the Pfaffian system is completely intsgrable in
the Probenius sense, i.e., defines a foliation.)

APPENOIX 1 -- CONNECTION THEORY
407
Set;
У*М - ^(y) + [^(υ),^(υϊϊ (1.2)
For χ € X, set
γ1 [χ) - У1 [γ) Ιχϊ . 11.3)
γ [χ) is a linear subspace of the tangent space that contains γ[χ). λβ χ
varies, we get a family χ ■* γ (χ) of tangent spaces. Let us say that γ is
one-regular if the die en β ion of these spaces is constant, in this case, γ
defines another Pfaffian system, which is called [by Cartan) the first derived
system.
Similarly, set:
y2[Y) - *αγ) + [^ιυϊ,^ιυϊ] + ivw »[^(υϊ, *ίυ)Π ιι.4)
γ2[χ) - ^2[γΜχϊ . 11.5)
Let us say that γ is too-regular if it is one-regular and if
dim γ2[Χ)
is constant as χ ranges over X. In this case, γ defines a Pfaffian
system called the aeoond derived вуз tern, continue in this way to define the
n-th derived system. The original system γ is said to be regular if all its
derived systems are regular. For simplicity, we shall consider only regular
systems. [One can prove that, in general, there is always an open subset of
X on which the system is regular. If the system is real analytic, this
subset is also dense in X, and its complement is contained in analytic
varieties of lower dimension.)
For χ С X, we have an increasing sequence of tangent subspacest
γ[χ) с γ1 [χ) cY2[x) ... [1.6)
Then γ ,γ ,... also define vector bundles over X. These are the
pull-back of the standard vector bundles over the Grassmann manfold. For
convenience, wa will make no notational distinction between the γ'β as cross-
section* of the Crasomann bundles and as voctor bund loo over X.
Set:
ΕΧ[χ) - γ1 [я)/γ [χ)
Ε2[χ) - γ2[χ)/γ1[χ) [1.7)

408 APPENDIX 1 -- CONNECTION THEORY
Each Ε (χ), Ε (x),... is a linear vector space. As χ varies, they define
vector bundles Ε ,E ,... over x. They also obviously play a basic role in
the study of the structure of Pfaffian systems.
He shall now define structure tensors, τ ,τ ,... tensor fields aaeoci-
12 12
ated with the vector bundles defined by γ ,γ ,... and Ε ,Ε ,...
To define τ , again work at a point χ of X. Pick ν ,v € Jf(Y).
Then,
[V1,V2](x)
is a tangent vector to X, which lies in γ (x). Consider its projection
mod γ(χ), i.e., as a vector in Ε (χ). We obtain a skew-symmetric map
(V1,V2) ■+ t1(V1,V2) .
Note now that τ only depends on the values of V and V. at x, not on
their derivatives, τ thus defines a skew-symmetric, bilinear map
τ1(χϊ. γ1 (χ) χ γ1 (χ) ■* Ε1 (χ) .
Explicitly,
t1(x)(V1(x),V2(x)) = Ινχ·ν ](χ) mod Υ(χ) . (1.8)
λβ χ varies, we obtain a bilinear vector bundle map
Τ : Υ4 +Ε . (1.9)
This is called the first integrability tensor. Notice that, by the vary
definition of Ε , τ is onto, hence it is sero if and only if the Pfaffian
system with which we began is Frobenius integrable.
We can now continue. For ν.,ν ,V € *^Y), consider the triple
commutator:
[V1,[V2,V3]](x) .
It lies in γ (χ). Its nontsneorial component is eliminated by projecting
mod γ (x). We thus obtain a tri linear map
2 2 2
τ (χ): γ(χ) χ γ(χϊ χ γ(χϊ γ (χ)/γ(χ) = Ε (χ)
Now, as χ varies, we obtain a tensor field τ as a trilinear bundle map
γχγχγ +Ε
This procedure can obviously be iterated to obtain n-th degree structure tensors
η _n
Τ:γχ···χγ ■+ Ε

APPENDIX 1 — CONNECTION THEORY
409
Ne can now discuss In what sense these tensors are "invariants" for the
equivalence problem. Let
Yi X+ <Λτ(ΧΪ)
γ': Χ' ■+ <^(Τ(Χ')Ϊ
define Pfaffian systems (of the same dimension) on manifolds X and X'.
Definition, γ and γ' are strongly equivalent {in the sense of Lie and
Cartan) if there is a diffeomorphism
φι Χ Χ'
such that
Y' (♦(«)) - v#(y(x)) (1.10)
for all χ С X .
In words, the natural actions of φ on the tangent bundle and on the
associated Grassmaim bundles intertwine the cross-section maps γ and γ*.
If + satisfies (1.10), it also satisfies
♦4(r(Y')) - ΊΚ(γ') . (1.11)
From (1.11) it follows that + acta on the derived systems Υ ,γ ,..., the
associated vector bundles, and intertwines the action of the structure
tensors. 9» "algebraic invariants" of these structure tensors will be
"equivalence invariants" of the Pfaffian systems.
Remark. "Иеак equivalence" of Pfaf fian systems means that thay car be
prolonged to be strongly equivalent.
Let Υ be another manifold and let
+ t Υ ■* X
be a submanifoLd map.
Definition, φ is a molutt&n of the Pfaffian system if the following condition
is satisfied!
♦•<Yy> c Y(v(y>> (1.12)
for all у С Υ

410
APPENDIX 1 ~ CONNECTION THEORY
Conditions (1.12) are often more conveniently viewed in terms of
differential forms. An exterior differential eye ten Λ in X is a collection of
differential forma satisfying the following conditions)
If θ ,θ С eT, then
If β ε t, and θ is in arbitrary form* then
β1Αβ2 € * *
i.e.. A* is an ideal in the Grassmann algebra.
If 6E /, then
d6 ε «T ,
i.e., «T ia a differential ideal.
A Pfaffian system γ,
Y!X* (^(T(X))
defines an exterior differential system β?(γ). Consider the one-differential
forms θ on X such that:
β(γ(χ)) - 0 (1.13)
for all χ ε Χ
(These are the Pfaffian forme dual to the vector fields Τ^(γϊ.) Α*(γ) is
then defined as the вшаlieat differential ideal of forms containing these dual
one-forms.
Definition. A submanifold map + s Υ -» X is a solution of the exterior
differential system A* if the following condition is satisfied:
♦*(β) - 0 (1.14)
for all θ ε *
Remark. The most accessible treatment in modern literature of Garten's
existence theorem for exterior differential systems [θ ] (completed by
KXhler ) is in Dieudonne's treatise [11].
If φ: Υ ■*■ x is such a solution manifold, one can define a set of
linear differential equations defined along the submanifold φ(Υ). Let

APPENDIX 1 - CONNECTION THEORY 411
t -»■ φ , 0 < t < 1
be a one-parameter family of submanifold mappings. For у € Y, let
v(y) - tangent vector to the curve t ■*■ φ (у) at t ■ 0
vi у ■* v(y) is a map Υ ■* T(X). Then, if θ is a differential font on X,
!?♦;«·>
(y) - Φ*(ν 4Θ) + d>*(vJ Θ) (1.15)
t-0
Thus, if φ is a solution submanifold for each t (or to "first order in t"),
then ν satisfies:
♦*(vj d6) + d>*(yJ Θ) - 0 (1.16)
for all β Ε /
These are (as linear differential equations for v) the linear variational
equations of the exterior system «T and the solution φ.
One can also look at this from the Lie point of view. If V is a vector
field on X, then» for β Ε /,
SB (β) - d(VJ β) + VJ d6 . (1.17)
Ίο solve the linear variational equations means to look for vector fields V
such that:
Φ·(^ν(β)) - 0 (1.18)
dφ*(VJ β) + φ·0Ν d6) (1.19)
Let Ν(φ) be the normal vector bundle to the submanifold N. It is
defined as the real ordered pairs
(y.v) , (1.20)
where у £ ϊ, and ν is an element of the vector space
Χφ(ν)/Φ·(ν ' (1*21)
which is, geometrically, the space of normal tangent vectors to the submanifold
φ(Υ)

412
APPENDIX 1 -- CONNECTION THEORY
(As usual In linear algebra when there is no quadratic form available to
identify an "orthogonal complement" to a linear aubspaoe of a vector apace»
the "normal vectors" are identified with the quotient vector space.)
It is readily verified that these equations only depend on the values V
taken on the submanifold φ(Υ) and are the projection of V in the normal
bundle Μ(φ) . Thai, the equations (1.16) determine a act of linear
differential equations for cross-sections of the vector bundle Η(φ).
2. FOLIATIONS, THE BHRESHAHH-REBB HOLOHONY GROUP, AMD THE BOTT CONNECTION
Let γ: X ■+ G*(T(X)) be a Pfaffian system, γ is said to define a
foliation for X ("involutive distribution", in Chevalley's language)
if the first structure tensor vanishes. This means that
Ι^ίγϊ,Τ'ίγϊ] € *αγ) , (2.1)
i.e., -y(Y) is a Lie subalgebra of У(Х).
The Frobenius Complete Integrability Theorem (in ite modern "global"
version, proved by Chevalley) assertβ that, for an γ satisfying (2.1)
and for each χ С X, there is a unique maximal connected solution eubmanifold
of X of dimension m, called the leaf of the foliation through ж.
Let
*t Υ -»- X
be such a leaf. Let V С V(X) and θ € 9 (X) be a vector field and a
Pfaffian form on X such that
βί^(γϊ) - 0 , (2.2)
i.e., θ belongs to the dual Pfaffian system. Then,
φ*(β) - 0 .
The linear variational equations (1.19) are then:
♦*(VJ d6) + d**(VJ Θ) - 0 . (2.3)
But, for И € У(у),
(VJ d6)(W) - d6(V,W)
- ν(β(Η)) - Η(β(Υ)) - β([ν#Η])
- -Η(β(ν)) - βίίν,Η])
(2.4)

APPENDIX 1 -- CONNECTION THEORY
413
Theorem 2.1. There is a flat linear connection on the normal vector bundle
Μ (φ) to a leaf ♦» Y+X of a foliation γ on the manifold X. The flat
cross-sections of the connection are than the solutions of the linear
variational equations of the underlying Pfaffian system.
Proof. One obtains cross-sections of NO) by the following geometric
process ι Take vector fields V on X, restrict it to φ(Υ), then project it
to the quotient vector space of the tangent bundle T(X) by the image
♦#(T(Y)) of the tangent bundle to Y, i.e., the tangent vectors to the
submanifold φ(Υ). Denote this cross-section by v. Let If be a vector
field tangent to φ(Υ). Set:
Vν - projection of [W,V] in the normal bundle (2.5)
It is readily verified that this operator defines a bona fide linear connection
in Ν(φ). (He use the Koszul definition of connection in vector bundles.)
It is readily seen that the flatness of the connection, in other words, the
vanishing of the curvature tensor, follows from the fact that *ν(γ) forme
a Lie subalgebra of T^(X), i.e., that the Pfaffian system Υ is "completely
integrable".
As for any linear connection, one can define the holomony group of this
connection on Ν(φ). It is a group of linear transformations on the fiber of
Ν(φ) above one fixed point. Since the connection is flat, the holomony group
is discrete. It is the obstruction to the leaf space of the foliation being
a manifold, in the following senset
Theorem 2.2. If there is a manifold Ζ and a submersion map is X -*■ Ζ
whose fibers are the leaves of the foliation, then the holomony group of the
foliation is the identity.
If the holomony group of the foliation is not the identity, it plays a
role in the study of the stability properties of the leaves, i.e., in the
sense in which leaves which come close at one point remain (or do not) live at
other pointe. Reeb's monograph [34] is the classical source for this circle
of ideas.
3. CONNECTIONS IN FIBER SPACES
Let X and Ζ be manifolds. Let
dim X - η
din z - m
w: X ■* Ζ

414
APPENDIX 1 - CONNECTION THEORY
be a eubmraion map from X to z, i.e., а шар
w#: T(X) ■* T(Z)
on tangent vectors is onto. Thus* by the implicit function theorem, X can
locally be parameterized by:
(y,z)
у € Rn~B , ζ € r" ,
with
w(y,z) - ζ . (3.1)
The geometric object (Χ,Ζ,τ) will be called a fiber шраал.
For each ζ € Ζ, the inverse image
w-1(z) = X(z)
(called the fiber above z) is (again* by the implicit function theorem) a
submanifold of X. It is* in fact, a regularly imbedded eubmanifold, i.e.,
its intrinsic topology as a manifold is the topology induced as a subset of
X. The tangent vectors to X(z) at χ С X(z) are the vectors ν € т(Х)
such that:
w#(v) - 0 .
These vectors are called the vertical veotore.
A vector field V € 'V(X) is said to be protectable under τ if there
is a vector field V on Ζ such that
w#(V(x)) - V (w(x)) (3.2)
for all χ С X .
Alternate geometric ways of putting this condition are important for
many geometric and physical applications. Here are a few:
»*(^ν, (β)) - Я'у(ж*(в)) (3.3)
for θ a differential form on Ζ .
If V generates a one-paremeter group t ■* exp(tv), i.e.,
if V is a complete vector field of diffeomorphisms of X» ., ..
for each t € R, the diffeoaorphism exp(tv) maps each
fiber of τ into another fiber.

APPENDIX 1 - CONNECTION THEORY
415
If ty#z) i« * local coordinate system for X satisfying
(3.1), then V and V are of the following font:
V - a(y,z> jpbUI L· .
V - b(z) |~
Let us state formally some properties of the projactable vector fields
that play a role in later sections. (The proofs are routine.)
Theorem 3.1. For each vector field V € У(Х), there is at most one vector
field V С У(г') such that
w#(V) - V ,
i.e., w# is a well-defined map; T^(X) -*- ψ{Ζχ).
Let
(τ>
be the set of all these projectable pairs (V#V)· i.e., f(w) is a subset
of the direct sum Lie algebra У(Х) φ У (Ζ) . Than:
a) 9(w) is a Lie subalgebra of У(Х) φ ΤΠΖ).
b) Τ*ι* map V -*■ w#(V) is a Li* algebra hOBOooxphism of 9(w> onto a
Lie subalgebra of y(Z) . The map
iv, v1) ■* ν
is a Lie algebra isomorphism of 9(») with a Lie subalgebra of У(Х) .
This Lie subalgebra of У{Х) is called the Lie algebra of vector fields
which are protectable under w.
The Lie algebra 900 is the basic algebraic structure assocated with
a fiber space. These vector fields are also central to what the physiciste
call the theory of даидш fimlde. Accordingly, I will call 9(τ) the gauge
[Liel algebra of the fiber spaca.
A pair (φ,α) of diffeomorphisma
+ t X ■* X
at 2 ■* Ζ
is called a gauge transformation of the fiber space (X,Z,w) (alternate
mathematical term: fiber epaoe automorphism) if the following condition is satisfied!

416
APPENDIX 1 -- CONNECTION THEORY
U - « - (3.5)
Consider (φ,α) as an element of the direct product group
DIFF(X) * DIFF(Z)
In fact, this defines the sat of all gauge transformations as a aubgroupt
denoted as G(w), of
DIFF(X) * DIFF(Z) .
(It is defined locally by means of differential equations, hence is an "Infinite
Lie Group" in the sense of Lie, Cartan, and Spencer.)
Let у be a point of z, and let
Z(z) - w"1!*)
be the fiber of the fiber space τ above the point z. Let (φ,α) € G(w).
Then, using (3.5),
t+(X(s>> - ow(XU))
- oz ,
whence:
♦ (XU)) - X(oU>> . (3.6)
(3.6) means that φ maps fibers of τ (parameterized by points of Z) onto
fibers, the action of α tells us how φ affacte the parameters of these
fibers. Now, it is readily seen that the infinitesimal generator rector
fields of one-parameter subgroup* of G( ) are the vector fields in 9(τ).
Tims, 9(t) is, in a sense, tha "Lie algebra" of the "Lie group" G(w).
(Unfortunately, the relation between "Lie groups" and "Lie algebras" is not
as tight for the "infinite dimensional" ones as for the "finite dimensional".
Now, the fiber spaces that occur naturally in geometry or physics come
with special geometric structures on the fibers
{ X(s)i z€z}
of т. Thus, it is natural to focus attention on the subgroup of G(w)
consisting of the transformations which preserve this structure.
For example, a common situation is that where the fibers are vector
spaces, i.e., τ is a vector bundle. One is then interested in the subgroup
of tha (φ,α) such that
φ: X(s) * X(s)

APPENDIX 1 -- CONNECTION THEORY
417
is, for each ζ € ζ, a linear automorphism of th· vector space. Such
transformations are than automorphisms of the vector bundle, and play a basic role
in elementary particle physics. Lie group representation theory (via the
theory of "induced representations"), inverse scattering structures, etc.
Let
VE(x) - { v€ Xs w#(v) - 0 }
be th· vmvbtaal vector» at x, i.e., the vector which is tangent: at χ to
the fibers of The assignment χ ■* VE(x) defines a (completely intagrable)
Pfaffian system;
VE с у (χ)
denotes the cross-sections. VE has two sorts of algebraic structuress
a) VE is a Lie subalgebra of У(Х)
b) VE is an ^(X)-module.
Of course, the module and Lie algebra structure are not algebraically compatible,
•inc* VE i· not a Li· algebra over th· ring ^(X) . However, it i· * Li·
algebra over a certain subring of ДОХ), namely the subring
w*(0"(Z))
consisting of the pull-backs under τ of the functions on Z. Explicitly,
note thets
IVl,fV ' Vltf>V2 * rIvi'V2J
for V ,V € VE , f € £"(X)
The function on the right hand side prevents the Jacobi bracket from defining
a Lie algebra over the ring ДОХ) . However, if
f € w*(£-(Z)) ,
then
V^f) - 0 ,
since V is tangent to the fibers of w, and f is constant on the fibers.
Hence, we have
[ν1(£ν2] - f[v1,v2l
for f e w*(*"(z>) . v1#v2 e ve ,
which is precisely the condition thet VE is a Lie algebra over the ring

418 APPENDIX 1 - COHHECTION THEORY
w*(#"(Z)). It will turn out that this structure of Lie algebra-cver-the-ring
τ*(«Τ(Ζ)) is basic to both the theory of gauge fields and the theory of
connections.
Definition, λ connection for tha fiber space ν is a Pfaffian system
χ + H(x) с χ
on χ which is oomplenantary to VE, i.e., for each χ € Χ, χ is the
direct sun of the linear subepaces VE(x) and H(x) . Given such a connection,
let Jf denote the cross-sections of the vector bundle χ + H(x). Jt is
then an ^(X)-submodul· of V(X). ttien, each vector field V € У(Х> can
ba decomposed (by linear algebra) into a sua
V 4 V .
where, for each χ € X,
V* (χ) С VE(x)
V (χ) € Η(χ) .
The possibility of such a decomposition is, for fixed x, just a mat tar of
linear algebra. However, it is important to realize that V and Vм depend '
in а С Way on x, i.e., themselves define vector fields. Then, the
module V(X) is a direct aim of submoduless
*4x) - ve φ л"
Tbis is the algebraic viewpoint. It should ba supplemented with the
geometric. Let ζ ba a point of Z, and let χ € X(z) ba a point in the
fiber above z. Think, of X as lying above Z, as follows:
X

APPENDIX 1 - CONNECTION THEORY 419
Think of the fibers X(x) as curves in X.
The tangent vectors, i.e., elements of VE(x), are tangent to these curves
H(x) nay be thought of as the space of values of J? at x. It is a
complementary Bubspac· to VE (x).
y^
Since VE(x) is the kernel of #, i.e., the fiber of "to the first order",
t maps vectors in H(x) in a one-one way. Bhresmann called them a set of
horiMontal vectors at χ (whence, the notation Η and Jt). Thus, as χ
varies, we obtain a family of tangent subspacss to X that representa the
geometry of the base Ζ "to first order". This is Ehresmann's intuition for
what Cartan meant by "the method of the moving frame".
Consider such an Jt as fixed. Let us bring in the gauge algebras 9(τ),
i.e., the set of pairs (V,V) of vector fields,
V € Tax) ,
V' € V{Z) ,
such thatι
w#(v(x)) - v'(w(x))
for all χ С X .

420
APPENDIX 1 - CONNECTION THEORY
For this section, it is convenient to identify (ν,ν'ϊ with v, i.e., to
identify 9(w) with a Lie subalgebra of V(x) . Sett
«(w^JT) - «(*> П ЛГ (3.7)
Theorem 3.1. Consider w# as a Lie algebra homomorphism of tf(w) ■+· ψ{%) .
The kernel is than VE, the Lie algebra of vectoral vector field·. w# la
then one-one on the linear subspace β(*,ΛΠ · 9(т) is a direct sum (as a
vector space) of VE and the linear subspace 9(t,JP).
Proof. Since H(x) Л VE(x) - (0), for all x, we havet
ЛГП VE - (0) (3.S)
We have already proved that V(X) is a direct sum (on a vector space) of VB
and JT\ That
V(t) - VE © «(т^Г) (3.9)
follows.
Theorem 3.2. w#, considered as a linear mapi 9(t#JP) ■* V(Z) is an
iaookorphlam.
Proof. We have just seen that it is one-one. lb prove that it is onto,
let v' С (Ζ). One can construct a cross-section Vi i* v(x) of the
vector bundle χ -»■ H(x) of the horizontal tangent vectors as follows!
For χ € X, v(x) is the unique element of H(x) which
goes into v*(w(x)) under the isomorphism w#i H(x) -*■%...
To finish the proof, we have only to prove that ν ie smooth, i.e., defines
a (C ) vector field on X. This follows from tha implicit function theorem
(since τ is a submersion).
Let
h: 1K(Z) + JTc ТИХ) (3.10)
be the inverse of w# restricted to 9(v,Jt*) · The connection is clearly
determined by h.
Theorem 3.3. Consider both V(Z) and 9(т) as AT(Z)-modules. Than, h
is AT(Z) -linear.
Proof. The action of 0{Z) on V(Z) is just the natural multiplication
of vector fields on Ζ by functions on Ζ

APPENDIX 1 - COHNECTION THEORY
421
(f#V) * fV: ζ f(z)V(z) (3.11)
bet ДОЕ) act on tf(T) a· follow·ι
Embed #"(Z) on ^(X) as the subring w#(^(Z)).
Then, let w*(^(Z)) act on φ(τ) by multiplication,
as in (3.11), i.e.,
fv - w*(f)V . (3.12)
To verify that h is ^"(Z)-linear is now evident from the fact that 0{Z)
is identified with functions on X which are constant on the fibers.
We now discuss oomplet*nsa$ of connections. Recall that a vector field
ν on a Manifold И is said to ba oorrpltte 1С it is the infiniteeimal
generator of a one-parameter group of diffeomorphlame on M. We can then denote the
one-parameter group as
t ■+ exp(tV) , — < t < ■
Then, if β Is a differential fore on M, we tuvei
1^- exp(tV)*(e) - SPy exp(tV)*(ej (3.13)
then, in the real analytic case, exp(tv) is given by Lie eeriest
t2 2
e*p(tV)*(9) - θ + tSey(B) + jj- #у(в) + "" . (3.14)
Definition. The connection h is said to ь· horizontally oomplete if the
following condition is satisfied:
For each horizontally complete vector field V on z,
h(V) is also horinzontelly complete.
Remark. Ehresmann included "horizontal completeness" in his definition of
"connection".
The horisontal completeness can ba used to prove some useful global
properties of the fiber spaces that carry them. I shall state a few of them
without proof here.
Theorem 3.4. If the fiber spacs w: X ■* Ζ admits a horizontally complete
connection, then it is a "local product" fiber space, in the sense that each
point ζ С Ζ has an open neighborhood U that the fiber space ι (U) above
U is isomorphic to the product fiber space U*t" (z).

422
APPENDIX 1 — COHNEaiON THEORY
Theorem 3*5. Suppose that X and Ζ are manifold* of the same dimension.
Let ι; X ■* Ζ be a fiber space map, which, in this case, is just a map with
everywhere nonzero Jacobian, i.e., a local diffeomorphiam. There is only one
connection, namely:
JT - (0) .
It is than horizontally complete if and only if τ is a covering map.
4. CURVATURE OF CONNECTIONS IN FIBER SPACES
Let X and Ζ be manifolds, n X>Z a fiber space map. ·(*) is
the Lie subalgebra of T"(X) consisting of the vector fields which are
projectable under w. 9(т) is an ^(Z)-module (but not an #"(Z)-Lie
algebra) . The vertical vector fields VE form a Lie ideal and an ДОХ)-
■ubmodule of 9(τ) (as well as an ^"(Z)-Lie algebra), λ ооппФ&Ьит will
be defined as an ^(Z)-linear mapping
h: V(Z) ■+ »(w) ,
such that
9(t) is a direct sum (as an ^(Z)-module) of VE and
ΜΤ^ίΖ)).
w#(h(V)) - V (4.2)
for V € ΐαζ)
Now, h is a map from one Lie algebra to another. However, it is not
necessarily a Lie algebra homomorphism. We can construct, algebraically,
an object which measures the deviation of h from a Lie algebra homomorphism.
Set:
Q(V1,V2} " IhtVi)'htV2)1 ' hiIVl(V2]) . (4.3)
Theorem 4.1. 8, defined by formula (4.3), is an #"(1)-bilinear mapt
ΐαζ) * Ψ\Ζ) ■+ VE
Proof. First, let us prove that the vector field on the right hand side
of (4.2) belongs to VE, i.e., is tangent to fibers of w. Recall that all
vector fields in 9(т) are pro jec table under w, and that tha map V ■* w#V
on projectable vector fields preserves Jacob! brackets. Hence,

APPENDIX 1 — CONNECTION THEORY 423
w#(Q(V fV >) - , using (4.2) and the right hand side of (4.3),
(w^ihiV^J, w,h(V2)] - [V1(V2]
whence,
RtVl'V € W *
It is obvious from formula (4.3) that V ,V2 ■* R(V ,V2) is R-bilinear
and skew-symmetric. He must prove that it is #"(Z)-bilinear. Let f С aT(Z) .
Then,
R(ev1(v2) - [hiev^hVj)] -hdw^v^)
- Iw*(f)hiv1),h(v2)] - h(fiv1,v2]-v2(f)v1)
- n*(f)[h(v ),h(v2>] - h(v2)(w*(f))h(v1>
- w*(f)h[vlfv ] - w*(v2(f))h(v1)
- ^(ίϊίΙΜν^,Μν^Ι -h((v1,v2D) +0
- w*(f)R(v1,v2)
This is what one mans by j9'(Z)-bilinearity, since j?(Z) acts on β(Ό
by multiplication by elements of **(#"(Z)).
The £4Ζ) -bilinearity of Ω, as expressed by Theorem 4.1, indicates
its tensorial nature. It is called the curvature tensora.
Now, h determines a non-iingular Pfaffian system Η on X.
H(x) - MT^ZJHx) (4.4)
for χ ε χ
Elements of H(x) are called horinontal vector a.
Theorem 4.2. The curvature tensor Ω vanishes if and only if the pfaffian
system Η defined by (4.4) is completely integrable.
Proof, Q zero implies (using (4.3)) that MY'(Z)) is a Lie subalgabra
of T"(X), whence, using (4.4), that Η is completely integrable.
Conversely, suppose that Η is completely integrable. Let
ЛГ - { V€ y(X)i V(x) € (H(x))f for xCx}
Then, "complete integrability" means that

424 APPENDIX 1 - CONNECTION THEORY
[Jf.Jt] с JT . (4.5)
We can then (using the Frobenius Theorem) find a local coordinate system
for X consisting of functions
ζ , у* (4.6)
1 <_ i, j !■ - dim Ζ
B + l < e,b < η - dim X ,
and associated "flat" vector fields ι
iz
(4.7)
·· ■$
d»1» ) - 0
dya»i> - о
dziO ) - 6*
dya(3b) - <ζ
such that the following conditions are satisfied:
χ are constant· on the fibers of τ, i.e.* are the
pull-back of a coordinate system on the base Ζ ,
dy*UT) - 0 . (4.9)
Let v,V ba vector fields on Z. Iben h(v), h(VM can ba written
in terms of these coordinates!
h(V) - Ai3. + A*3
(4.10)
MV) - Bi3. + B*3
X A
The conditions that:
w#(h(V)) - V ,
(4.11)
w#(h(V)) - V ,

APPENDIX 1 - CONNECTION THEORY
425
enforces the following condition:
MvXx1) - a1
and
hlv'JliS - в1
are the pull-backs of functions on Ζ
"Леве conditions require that
3ft(Ai) -
-
(4.9) requires that
a' - о
Hence,
0
3a(Bi)
- Bb
(4.12)
(4.13)
h(V) - w*(dsi(V))Ai (4.14)
for each vector field ν on Ζ .
(To make sense of this formula, the de should ba interpreted as a form
on Z.) Thus,
0(vfV) - h([v,V]) - [h(v), h(v')]
- h([A13i, Bj3 ]) - [Ai3i, Bj3J (4.15)
But, using (4.14), we see thatt
h([Al3i, Bj3 ]) - [Ai3if Bj3 ] ,
which proves that
Q(v_,v2> - 0 .
since vi'V2 ·*"· arbitrary vector fields on Z, it follows that
a - 0 , (4.16)
as required to finish the proof. Q.E.D.
Remark. This proof looks like a cheat, since (4.16) follows from (4.15)
somewhat trivially. The point is that the Frobenius theorem has been used
in a non-trivial way to choooo a special coordinate system in which the
h-mapping take· the special "flat" form (4.14).

426
APPENDIX 1 - CONNEaiON THEORY
Locally, we see that curvature zero connections are not very interesting.
However, from the global point of view* this is not at all so, especially for
physics, because both the "isomonodroay" and "isospectral" deformations that
occur in aoLiton-nonlinear wave theory involve curvature aero connections.
The following result (of Ehresmann) is one of the simplsst global theorems.
Theorem 4.3. Let w: X -*· Ζ be a fiber space with a hor iron tally complete
connection. The horizontal vector fields Jft then define a foliation of X.
Let ♦: Z' -*- X be a leaf of this foliation. Then, the composite map
w$: Z' ■+ Ζ
is a covering map. In particular, if Ζ is simply connected, then τφ is a
diffeomorphism, and the fiber space is isomorphic to a product
Zx Υ
where Υ is the fiber above one point.
We now describe curvature in terms of differential forma. Let w: X ■+ Ζ
continue as a fiber space with a connection defined by an ^(Z)-linear mapping,
a horizontal lifting, of vector fields on Ζ to vector fielde on X. We have
defined the curvature form ae an ДО2)-bilinear mapping from vector fields
on Ζ to the vertical vector fielde on X. Q has soma of the algebraic
flavor of a two-differential form. In this section I will discuss how it can
be related more precisely to two-differential forms on Z.
First, let us review material about differential forms. Let Μ be a
manifold. A {eaalar-valued) two-form on Μ is an ^"(M)-bilinear skew-
eymmetric map
ω: У(М) x T^M) ■+ ^(M)
Its value at ρ £ Μ is an R-bilinear, skew-symmetric map:
ω(ρ): Μ xM ■+ R
Ρ Ρ
Then, u can be identified with the tensor field
Ρ ■* ω (ρ)
To put this another way, let Б be the product bundle
Ε ■ MXR
over M. Let E(p) (isomorphic to R, of course) be the fiber above the
point p. Then, u(p) is an R-bilinear skew-symmetric map
ω(ρ): Μ * Μ ■+ E(p)
Ρ Ρ

APPENDIX 1 — CONNEQION THEORY
427
Μ is the fiber of the tangent bundle, u can than be defined aa a bilinear,
skew-symmetric bundle вар
ω: T(M) xT(M) ■+ Ε . (4.17)
We can now generalise» replacing the product bundle Ε with an arbitrary
vector bundle over M. this leads to the concept of a differential form on a
manifold Μ with values in a vector bundle E. An alternate algebraic
version would be an #"(M)-bilinear, skew-symetrie вар
ω: У(М) χ Там) ■+ Γ(Ε)
between the nodules of aross-eeotions of the vector bundles. Geometrically
(and physicall) all these concepts are equivalent. Of course, in various
applications one formulation is wore natural and/or convenient. If one wants
to calculate in a coordinate free way, the module approach is usually more
usefulf for calculations in local coordinate systems, the vector bundle
approach combined with local product structures for the bector bundles is
usually better.
Let us now return to a general fiber space we X ■+ Ζ with connection:
η: ΤαΖ) +JTC «(τ)
and curvature:
Q: T^tZ) x y(Z) ■+ VE .
We have seen that Ω is ^(Z)-bilinear and skew-symmetric. Thus, if there
were a vector bundle Ε over Ζ such that its space of cross-sections Γ(Ε)
was isomorphic (as a ДОZ)-module) to the ДОZ)-module VE of vertical
vector fielde, then Ω would be a too-differential form on Ζ irith values
in E.
Now, there certainly is such a bundle, but its fiber is infinite
dimensional, namely,
Ε - {(x,Wb xEz, ИЕЯГ1!!))) (4.1β)
in worde, the fiber above a point ζ С Ζ is the vector space of smooth vector
fields on the fiber w~ (x) с χ of τ above the point x. (Since τ is a
submersion вар, note that τ (ζ) is a regularly embedded submanifold of X,
hence, this vector space is well-defined.) Most of the examples encountered
in geometry and physics involve the following special assumptions:
There is a subbundle E' of Ε with finite dimensional .. .».
fiber so that й takes values in E'.

428
APPENDIX 1 -- CONNEQION THEORY
We have seen one example of this ι For the flat connection* E' can be taken
as the zero bundle. The standard sort of principal bundle ооппесЫопа offer
another class of examples satisfying (4.19). In this case, E' can be taken
as a vector bundle whose fiber is a finite dimensional Lie algebra Ж, which
is the Lie algebra of the structure group К on connections. A special
example of this sort is the affine connection, i.e., linear connections in
the tangent bundle. The curvature takes values in the vector bundle over Ζ
whose fiber is the space of linear maps of the tangent bundle to itaelf.
He will now review the theory of connections in principal fiber bundles.
They play a particularly important role in both mathematical physics (where
they are called Tang-Mills fields) and differential geometry.
5. CONNECTIONS IN PRINCIPAL FIBER BUNDLES
Let ii X+ Ζ continue as a fiber space. Let К be a Lie group which
acta on X as a transformation group. Suppose this action is free, i.e.,
the following condition is satisfied:
If kx - x, for к £ Κ, ζ С Xr then к - identity. (5.1)
Suppose also that
К acta transitively on the fiber of i. (5.2)
With these conditions, (Χ,Ζ,τ,Χ) is called a principal fiber bundle with К
as structure group.
(5.1) and (5.2) imply that each fiber can be identified with К itaelfi
but not in a unique way. This suggests one way to construct such bundles,
as products Ζ χ Κ:
X - Zx К
k(z,k') - (z,kk') (5.3)
for ζ £ Ζ , к,к* С К
w(z,k) - ζ
One can also construct another bundle, by letting К act on the
X' - ZxK
(5.4)
k(z,k') - (z,k'k_1)
w(z,k) - ζ .

APPENDIX 1 - СОНИЕПЮН THEORY 429
These bundles are isomorphic. The isomorphism is
U,k) -* U,k-1) <*■*)
Here is a main geometric property of this type of bundle.
Theorem 5.1. Let и X -*· Ζ be a principal fiber bundle with К as structure
group, i.e., let (5.1) and (5.2) be satisfied. The cross-sections Г(т) are
then in one-one correspondence with the isomorphisms of X with the product
bundle (5.3).
Proof. Let
γ: ζ -»· χ
be a cross-section. Define a map
φ : ZxK -t- X
Υ
4 (z,k«) - k'yU) . (5.6)
Then,
4 Ck(z,k')) - 4 (kiMc1))
Υ Υ
- kk'YU)
- kf U,k') ,
i.e.,
4 intertwines the aetion of К.
Υ
It is readily seen that 4 is a diffeomorphism. Q.E.D.
Let (X/ZrT/K) be such a principal bundle with К as structure group.
The Lie algebra JT acts on s Lie algebra of vector fields on X. It acts
freely, i.e., if
V(x) - 0, for V £ Jt, χ £ X, then V - 0 . (5.7)
Since
wOui) - w(x)
for к С κ ,
we see that:

430
APPENDIX 1 -- CONNEQION THEORY
(5.8)
for ν ε дг
i.e.,
JTC VE , (5.9)
where VE Is the Lie algebra of vertical vector fields. One can then define
a linear map
ДОХ) ® Jt -► VE (5.10)
as follows:
f ® V ■+ tV . (5.11)
Theorem 5.2. The map (5.11) is an isomorphism between the ДОХ)-module
ДОХ)® J? and the ДОХ)-module of vertical vector fields for the principal
bundle.
Proof. Let
vi v-
be a basis for Jf. For each χ С X,
V.Wm.wVU)
ι η
is a basis for the vertical vectors. Thus, any vertical vector field V can
be written as
V - f1(x)V1(x) + ■■■ +fB(x)Va(x) (5.12)
Since the vector field* V.»...,V on χ are linearly Independent, the
coefficients χ ■+ f, (x),... ,f (x) are С functions, as χ ranges over X.
ι η
Q.E.D.
Of course, the map (5.11) is not a Lie algebra isomorphism between the
group algebra ДОг) ®ЛГ and ТЕ. However, consider T'tjptX)) Φ Jf as a
Lie subalgebra or the gauge (Lie) algebra ДОх)®«Ж.
Theorem 5. 3. The map (5.12) is a one-one Lie algebra homomorphism from the
Lie algebra
тг*(ДОг)) ®JT (5.13)
to the Lie algebra VE. In other words, the Li· algebra VE has a Lie
subalgebra isomorphic to the Lie algebra (5.13).

APPENDIX 1 -- CONNECTION THEORY
431
Proof. Obvious.
Now, let hi consider connection· for the principal bundle
(Χ,Ζ,τ,Κ) .
Let
H: X -► H(x) С X
X
be a field of horizontal subspaces for w. Then, for each χ С X,
Χ - JT(x) Φ Η , (5.14)
χ χ
where ЛЧх) denotes the set of values at χ of the vector fields in &.
One can thus define a Jtf-valueci one-form 8 on X by the following formula ι
8(v) - element of Ж such that v-fl(v)EH . (5.15)
Thus,
Hence,
8(v) - 0 if and only if ν С Н (5.16)
8(h(V)) - 0 (5.17)
for V £ ΤαΖ) .
τηβ connection can be defined by the jr-valued one-form Θ. of course/
we have the alternative of defining tha connection as an ^"(Z)-linear map
h: f(Z) ■+ »(w) (5.18)
We are now in position to derive what Cartan called the Structure
Equatione for the connection. Cartan did thie by uein? baeee for the
differential forms on X (which he called "moving frame·") conveniently adapted to
the geometric situation. we shall do this also.
Let us choose indices in the following ranges and the summation convention
on these indicest
1 <_ i, j,k <_ m - dim Ζ
m+ 1 < a,b,c < η - dim X
1 < U/V,w < η .

432 APPENDIX 1 -- CONNECTION THEORY
Let <u ) be a basis for one-forms on X such that
ω - 0 defines the fibers of τ , (5.19)
i.e.,
ui(JT) - 0
(5.20)
u (AT) is constant
Theorem 5.4. If γ: Ζ -*· X is a cross-section of the fiber space (X,Z,w),
i.e., if
ιτγ - identity , (5.21)
then the one-forms
γ*(ω1(,...,γ*(ωΙΙΙϊ (5.22)
are a basis for one-forms on the base space Z.
Proof, litis is just linear algebra. Notice that the number of forms in
question is equal to the dimension of Z. Hence, it suffices to prove that
thay are linearly independent at each point of Z.
Suppose otherwise, i.e., that there is a point ζ С Ζ and nontext? real
numbers λ, such that;
λ1γ*(ω1)(ζ) - 0
or
or
Y*(Xiaii(s))
λ^Ιγ,ίΖ^Ϊ) - 0 . (5.23)
γ is a cross-section, i.e.,
1 - *γ .
Hence,
1 - ιτ#Ύ# .
It follows that the tangent space X is the direct sum of the subspaoes
-1 Ύ1*' i
w# (0) and γ#(Ζ ). It follows from (5.23) and the hypothesis that u is
zero on tha vectors that are tangent to the fibers of w, that

APPENDIX 1 -- CONNEQION THEORY 433
which implies that
Xi - 0 ,
since the one-forms u are, by hypothesis, linearly independent at each
point of X. Contradiction. Q.E.D.
Mow, we can exterior differentiate the forms
(u ) - (u ,u )
to obtain what Cartan called the Structure Equations of the Moving Frame ι
*," - fU »VA«U (5.24)
vw
fU + fU - 0 . (5.25)
vw wv
then (f ) are, of course, functions on X. However, soae of thee are
oonatant (and soae of these coordinates are пето) as a consequence of the
geoaatry.
First, the Pfaffian system:
ω - 0
is completely integrable, i.e., defines a foliation on Z. (Its leaves are
the fibers of т.) Hence,
<=L, - ° · (5·26)
Second, the u restricted to the fibers of τ are (when they are identified
with the Lie group K) just tha Maurer-Cartan forma of K, i.e.,
f. ■ constant
*° (5.27)
■ structure constants of the Lie group К .
Let us now rewrite (5.23), taking into account relation (5.26) and (5.27)ι
du1 - f* wJ a uk + f* ω* α ω* (5.28)
, a _a b с _a i j _a к b ._ _AV
dw - f. ы а ы + f. , ы a mj + f. u a u . (5.29)
AC 1] D
Mow, fix a cross-section map
γι Ζ -»· Χ .
By Theorem 5.4, the γ*(ω ) are a basis for one-forms on Z.

434
APPENDIX 1 - CONNEQION THEORY
Let (V ) be the dual basis for vector fields on Z,
ΎΜοΑν > - «* . (5.30)
The h(V ) are then horizontal vector fields on X» i.e.,
u*(h(V )) - 0 . (5.31)
The curvature tensor fl is determined by its values on tha verical vectors.
QtW " η(ΐνν} " l^tV±), h(V )] (5.32)
uiQCV^V )) - 0 (5.33)
Sett
Q* - u*(B(VifV )) . (5.34)
They are the oomponentB of the curvature tensor in this moving frame. Notice
that they are functions on X. They do depend on tha choice of cross-section
γ. (γ is, of course, the "moving frame".)
Theorem 5.5. The vector fields h(V,) ■ Vj on X are the dual to u , i.e.,
u*(V·) - 0 (5.35)
u3(V') - б| . (5.36)
Proof. These are just relations (5.30) and (5.31) rewritten.
For a point χ of X, to calculate fl.. (which is "tensorial"), we
can take advantage of the freedom of choice in tha moving frame. Namely,
we can, without loss in generality, suppose that the (u ) satisfyt
du)1 - 0 at χ (5.37)
(What this means is that, if (u ) do not satisfy (5.37), a new set (u )
can be chosen so that
dti) - 0 at χ
u.(x) - u (x)
This trick (which is a version of the method of "normal coordinates") often
dramatically simplifies computetion).

APPENDIX 1 - COHNEaiOH THEORY
43S
Let
ζ - w(x) .
(5.3Θ) implies that
Hence,
Hence,
[ν±,ν ](x) - О . (5.3β)
fl(Vi,Vj)(x) - - [h(Vi)f h(V)](x) (5.39)
- - [V|,V'](x) . (5.40)
flij " * "*t IV^V'l)
- , using (5.35) ,
da*(V'fV·)
- , using (5.26)
2f* . (5.41)
We have now proved:
Theorem 5.6. The structure relation (5.25)-(5.36) determine the curvature
tensors. Explicitly, we have:
. i .i j к _i а к ., ...
du - f, или +f.u α ω (5.42)
.a ,* Ь cl.ai j
du ■ f. ω лы + — fl , u л u
DC 2 1J
,a к b
+ r.. ω α ω
КО
(5.43)
Let us examine the third term on the right hand side of (5.43). Its
vanishing expresses an important property of the connection.
Note that tha Pfaffian equations
ω1 - 0
define the horizontal vectors, which, of courser determine tha connection.
Let V be a vector field on JT. It satisfies the relations

436 APPENDIX 1 -- CONNECTION THEORY
■*(*> - 0 (5.44)
w*(V) - conetent . (5.45)
Bene·,
i?v<«*) - d(VJ и*) ¥ V J dw* . (5.46)
But, th· first term on th· right hand side of (5.46) vanishes, because of
(5.45). Using also (5.43), we havei
#„<«*> S -f* wb(V)«k mod (и*) (5.47)
V КО
И* have prov*di
Theorem 5.7. The horixontal Pfaffian system Η is invariant under the action
of the structure group К if and only if the following relations are satisfied!
f*^ - 0 . (5.48)
Thus, in trying to simplify the structure relations for the connection
(following Cartan's technique) we encounter in a natural way the conditions
Ehraamenn imposed (12], namely, the Invar lance of tha horizontal Pfaffian
system under the structure group of the bundle. Let ue call these K-imOTlOKt
oomsoticna.
Recapitulation of the Basic Definition.. Let wt X ■+ S be в principal K-bundle,
with a Lie group К acting freely on X and transitively on the fibers of т.
A connection for the fiber space, defined by a field χ -*- Η (χ) of horixontal
tangent subspaces, is said to be VL-imxxriant if the following condition is
satisfiedι
к*(Их} " \x t5*49>
for all к € κ, χ € Ζ .
Ие can now sum up tha "moving frame" relations in the following, more
precise way, keeping account of tha global situation.
Theorem 5.8. Let (Χ,Ζ,τ,Κ) be a principal K-bundle, with a K-invariant
connection ( С a connected Lie group) defined by a field ж -*■ Η (χ) of
horizontal tangent subspaces. Let Ct( be the Lie algebra of K, realised
as a Lie algebra of vector fields on X. Let Ι ω ,ω*) be a basis of one-
forms on an open subset 0 of X satisfying tha following relations!
(i)*(JT) - constant (5.50)

APPENDIX 1 - CONNECTION THEORY 437
«*(H ) - 0 (5.51)
JTJ ω1 - 0 . (5.52)
Thus, there ar· differential forme in U
such that:
ίω1 - (ik α ω3 (5.53)
Ad* - Ω* + λ* ub a (dC (5.54)
be
with tha following conditions satisfied!
JTJ Ω* - 0 (5.55)
XL are the structure constants of the Lie
*° a
algebra Cf(, i.e., the f are constant (5.56)
DC
on tha fibers of *.
Proof. Start from relations (5.42)-(5.43). Set
ω* - - f* uk + f1 ω* (5.57)
Ω* - γ Ω* ω1 a uJ . (5.5β)
Note that (5.55) is satisfied using relation (5.52). To prove (5.56)
let (V ) be a basis for JfC. Then,
[V ,V. ) - Xе. V ,
a b ab с
trhare λ . are constants. (The structure constants of tha Lie algebra Jff.)
an
Hence,
du*(vv.»v^> " VK*w*tv^>> * ν^(ω*(ν. )) - bi*([V. rV ])
DC DC CD DC
- 0 - 0 - λ*
be
Q.E.D.
Of course, we see again from (5.54) why the vanishing of tha curvature
tensor, which is essentially Ω*, determines tha "flatness" of the connection,
i.e., the complete integrability of the field χ -*- Я of horizontal tangent
subspace (or, dually, the Pfaffian equations ω -0), which determine the
connection.

438
APPENDIX 1 - CONNECTION THEORY
6. LIE ALGEBRA-VALUED DIFFERENTIAL FORMS AND ТЛЕ HAORER-CARTAN EQUATIOHS
We have меп that Equations (5.54) are the basic structure equations of
a K-invariant connection on a principal fiber bundle with ΛΓ as structure
group. I now want to pause in the development of connection-gauge field-Yang-
Mi lis theory to review some relatively elementary and standard material in
differential geometry and Lie group theory associated with the "Maurer-Cartan"
equations.
Let X be a manifold. We have already defined vector-bundle valued
differential forms on X. In particular, vector-space valued forms are a
special case, where the vector bundle Ε is the product
Ε - X« V
of X with a vector space V. Of courittt such β. fona can be defined directly
as a multilinear mapping
Θ: T(X) χ ■■- к Т(Х) ■+ V
In connection theory we are interested in tha special case that
v - JT ,
tha Lie algebra of a connected Lie group K.
We have to be more specific about the definition of Ж. The most
natural algabraic-point-of-view choice is to say that it is defined as the
set of one-parameter subgroups of K. Thus, for V € Ж
t -*■ exp(tv)
is, by definition, the one-parameter subgroup of К essociated with it.
JfC acts freely in two ways on itself, by left and Tight translation:
Ι- 1*'ϊ - **'
(6.1)
R.Ot') - k'k"1 .
The following result (which we will not prove here) is, in a sense, the main
theorem of Lie group theory.
Theorem 6.1. Jf can be made into a Lie algebra
with the following properties:
a) There are Lie algebra isomorphisms

APPENDIX 1 - CONNECTION THEORY 439
oR: CK -*■ V(JT)
b) For V € C/C, α (ν) (»sp. α_(ν)) is th· infinitesimal
L R
generator of th· one-parameter group of diffeomorphisms
^Wttv) tr"p' t"Rexp(tv))
c) [oR(jr),,oL(jr)] - 0
d) If V € У(К) satisfies
[V, or(JT)] - 0 ,
then
ν ε c^tor) .
·) Relation (d) holds if L and R are permuted.
f) For each к £ K, the maps
V ■* oR(v) (k)
V -*■ a (v) (k)
L
are vector space isomorphisms of ЛС with the tangent space K.
to К at k.
g) If IV ), 1 <_· <n, is a basis of CfC, then
ae(V ) and a_(V )
κ a L· a
are both ^(K)-module bases of ТИК). (Such bases are called,
in differential geometry/ absolute parallelieme.) Thus, if
[V ,VJ - Xе. V , (6.2)
a b ab с
with, oonetant X (called the structure acme tan te of the Lie algebra
for tha given bases), then
■W' W1 - CVV t6-3>
■W- VV1 - XL°RtvC) ■ <6-4>

440
APPENDIX 1 -- CONNECTION THEORY
h) The dual basis (ω.) of one-forms on К to the a. (V )« i.e.,
l La
th· οηβ-fonu which satisfy the following relation:
uf(eTtV.)) - β* r (6.5)
L L D D
satisfy the following relations:
·*" " IXicULAUL (6·6>
Of coureer also
- a 1 ,a Ь с ., _.
% " 2XbcULAUL t6'7)
with
·£<«.<*.)> - β* .
R R b b
i) The (ы ) are invariant under left tranalatiorn
L
L*(uL) - ω" (6.8)
for к £ К
St taj> - 0 (6.9)
К
for ν € CfC
Equations (6.6) considered as a set of differential equations for tha
one-forms ω . are called tha Maurer-Cartan equations. They were Cartan'β
L
favorite tool in the development of differential geometry and Lie group theory.
Lie preferred vector fields, Cartan differential forms.
One confusing feature is that ы are generated using the right trans-
L
lation of К on itself.
Ne can now define ЛС-valued one-forms on K, called the Мешгег-Cartan
forme:
• if
L La
ω - щ V
R R a
(6.10)
These objects are easy to calculate if К is exhibited as a group of
matrices, say:
К с GL(N,R) .

APPENDIX 1 - CONNECTION THEORY
441
Then
k"1 dk (6.11)
dk k"1 . (6.12)
How, there is a notational convention inherent in these formulas. It is
necessary to identify J€ with a set of matrices, and to identify the tangent
bundle
T(K)
with
к « JT ,
i.e., a pair (krH) of а к £ К and an Ν* Ν matrix H. These
identifications are confusing, and it is doing these computations (which, of course,
are necessary in correlating the coordinate-free geometar's notation with the
physicist's) that the beginner most oftan gets confused. Bare are some
examples.
Example 1. К - 0(N,R), the N«N real orthogonal matrices. K is then
the set of Η € GL(N,R) such thet
MIT - 1 (6.13)
Τ
(Η is the transpose of H. 1 is the identity matrix.) Then,
0 - dHMT + M3MT . (6.14)
Since HT - h"1, (6.14) is:
dm'1 + (dHM*1)7 - 0 . (6.15)
Thus, if we define JT as the Lie algebra (under matrix commutation) of Ν χ Ν
skew-symmetric matrices, then
ωΗ - dHM*1 (6.16)
is the right-invariant Cartan-Maurer form. Relation (6.15) says that <nR
is a one-form on К with values in ЛГ, the vector spaced skew-symmetric
matrices.
All the other "classical" matrix groups can be handled by this technique.
We can obtain the equations satisfied by ω and ω by applying
L R
exterior derivatives to both aide* of (6.11) ι
aar - dik"1) a dk . (6.17)

442 APPENDIX 1 -- CONNECTION THEORY
Now,
О - d(kk )
- (dk)k"1 + kdtk"1) ,
dik"1) - -k"1dkk"1 . (6.18)
Combine (6.17) and (6.18)t
duT - k"Xdk a k-1dk
L
(6.19)
Now, in (6.19), th· symbol л denotas extarior product of matrioea of
differential forms. For example, in the 2*2 case
/Ull И12\
\ω21 ω22/
("" "").(■" "")
\«21 ω22/ \*21 *22/
/ ω12 Α ω21 ' ωΐ2 Α ω21 + ω12 Α ω22 \
^ω21 Αωΐ1+ω22 Αω21 ' ω21 Α ωΐ2 '
Let ца now intarpret (6.19} in tazns of abstract Lie algebra theory.
ω_ is a Jtf-valued one-form on K. For V ,V € •y(K),
L 12
dw_(V_,v,) - ν,(ωτ(ν,)) - ν,(ωτ(νι)) -ωΠν.,ν,]) (6.20)
L12 a L 2 2 L 1 1*12
(6.21)
«"bA-b,(W " WW - WW
- w· w1 ■
Bane·, (6.19) i· «equivalent to the following relationι
DftL - 0 (6.22)
L

APPENDIX 1 - CONNECTION THEORY 443
whereι
DuiL(Vl(V2) - νι(ωίν2)> - VjiuiV^) - ul(IV1,v2) - Ιω^ν^, uL(V2))
(6.23)
We can now abstract some useful definitions for Lie algebra-valued
differential fornsi
Definition. L»t JtT be ι Lie alg*br». X a manifold. Let
ui T(X) ■+ JfC
be a Jff-valued one-for» on X. Then, Eta is a Jff-valued two-for» on X,
defined by the following fornilai
DM(V ,V2) - V1(U(V2)) - V2(W(V1)) - u([V1#V2)) - [*{νχ). uiV^)] (6.24)
He now wrLta the Maurer-Cartan equations in terns of this operation.
Theorem 6.2. Let к be a Lie group and let (V*J be a basis of Jf. Let
(u ) be a dual basis of tha left (or right) invariant one-forms on K. Let
u - ω V* (6.25)
a
be the Jlf-valued left invariant one-for» on K. Then
Du - 0 , (6.26)
where D is the operation given by formula (6.24].
Proof. Since D , as given by (6.24), is obviously left-invariant, it
suffices to prove (6.26) at one point. To do this, it suffices to prove that
ΐΜν^,ν*3) - 0 . (6.27)
Using (6.24),
Εωίν^,ν*3) - ωίίΑν*3)) - MV*), ωίν*3))
- ω ([ν*,Λ)ν° - [ω (V*)VC, ω (V**)Vе)
с с с
- [Αν*] - ιν\ν*>
■ 2 '
which proves (6.26), hence also Theorem 6.2.

444 APPENDIX 1 - CONNEQION THEORY
Theorem 6.3. Let X be a manifold and let
+ i X ■+ К
be a map. Let
θι T(X) -*■ «
be the one-form which is pull-back of tha JT-valued Maurar-Cartan form и on
K, i.e.,
θ - τ*(ω) ,
where ω is given by (6. 10) . Then,
Όβ - 0 , (6.28)
4
where D is the operator given by (6.24).
Proof. Obviously, the D operation is "covariant*, i.e., preserved
under the pull-back map φ*.
Now, for the converse.
Theorem 6.4. Suppose X is a manifold with a Jtf-valued one-form θ on X
such that:
0Θ - 0 .
Then, each point χ € X has an open neighborhood U and a map
v u ■*κ
such that:
♦5Ы - θ , (6.30)
where ω is the (left-invariant ЛС-valued) Maurer-Cartan form on K.
Proof. Construct!
У - Χ" Κ . (6.31)
Consider the pfaffian system
θ - ω
on У. As a consequence of (6.28), it is completely integrable (see Remark
below). Consider a leaf L of this foliation. It is readily seen that tha
projection

APPENDIX 1 - CONNECTION THEORY
445
L ■♦ X
is a local diffeomorphism. Choosing a local Inverse
XCO ■♦ LCXxK ,
following by the projection
XxK-fK
provides the вар φ .
Remark. I will make this argument more explicit in case К is a matrix
group, since it is quit* important in the theory of nonlinear waves. Suppose
suppose
К С GL(N,R) ,
i.e., an element of К (or ЛГ) is an N*N real matrix. Let
(x1) , 1 £i,j £ η
be Cartesian coordinates on R . Then
θ - Ai dx1 ,
where A.(x),...,A (x) are N*N matrices, which are elements in ЛС.
ι η
Condi «ion (.6.29) isi
W " VV " IAi'V " t6'32>
We are looking for k (€ к ) as a function of x, so that
θ - k"1 dk , (6.33)
i.e.,
3i(k) - kAi . (6.34)
We see that (6.32) are tha integrability conditions for the differential
equations (6.34) . (k as a function of χ can be determined by solving
a set of ordinary, linear equations depending on parameters.)
For example, set
kit) - k(at) ,

446
APPENDIX 1 - CONNECTION THEORY
Then,
*}*- - aVtatJktt) (6.35)
at i
These linear equations can be solved for all t. One aees (using a clasaical
argument, which in modern terms ia eaaentially the Frobenius theorem) that
the solution* of (6.34) exiat over all of R . Thia ia eaaentially a
"monodromy" argument, depending on the aimple connectivity of R . we will
now generalize it.
Return to the caae of a general manifold X and a connected Lie group
K, and a Jlf-valued one-form on X, 6 such that
Όβ = d6 + Ιθ,θ) - 0 (6.36)
Then, X can be covered by open seta
iU) ,
that, for each U € (u), there ia a map
V " * K
such that:
♦Jiu) - θ , (6.37)
where ω is the Jlf-valued Maurer-Cartan form on K.
One can arrange things (general theorem about paracompact connected
manifolds) so that the covering {0} has the following property]
Whenever U,U" € {u} intersect, the intersection ., -_.
(b. jo j
U Π υ' is oonnected.
Then, one sees (since the solution φ of (6.37) is unique up to left
translation) that, on each U П υ',
♦u - "uirV · t6-M>
where (*.»..) ia an element of K. Whenever U,U',tT € iu} intaxsect
кш- - Wu · (6·40>
Condition (6.40) is a "Cech cocycle condition". If X is simply connected,
a standard argument gives the existence of
such that, whenever U Π U' is non-empty.
a standard argument gives the existence of a set {k } of elements of K,
U " kUU'kU· '

APPENDIX 1 -- CONNECTION THEORY
447
Thus,
«tfv - ♦
is globally defined on X', and gives a solution to the Haurer-Cartan equations
φ·(ω) - θ
If X is not simply connected, there is a subgroup of the fundamental group
π (X) of X which is the "obatruction" to the global existence of φ.
Let
(Χ,Ζ,π,Κ)
be a principal fiber bundle with К as structure group. Suppose
Η: χ ■♦ Η
χ
is a field of horizontal subspaces» i.e., a connection that is K-invariant.
Let Jt be the ^(X)-module of vector fields V such that:
V(x) € Η
for all χ Ε Χ
Let tf(π) be the set of vector fields V on X which are projectable
under ir. Let Jf be the Lie algebra of K, considered as a Lie algebra
of vector fields on X. VE, the vertical vector fields» is isomorphic
(as an ^(X)-module) to ^(X) ® Ж .
Suppose U is an open subset of X» with a basis
(ω ,ω ) , 1 <_ i» j <^ m ■ dim Ζ
m+ 1 <^ a,b <^ η
of one-forms such that:
The Pfaffian system
i
ω - 0
is completely intagrable and defines the fiber of π»
i.e.» the vertical vectors. ω - 0 defines the
horizontal vectors
d(Ua(jT)) - 0 .
As we have seen in Section 5» the following structure equations are then
satisfied:

448 APPENDIX 1 — CONNECTION THEORY
du - ω a uJ (6.42)
dw* - λ*^ ык а ыС + Qa , (0.43)
where (ω ) are one-forme on 0, the Q are two-for» and 0 such that:
JlfJ 0* - 0 . (6.44)
Theorem 6.5. Let (V ) be the basis of Jf such thatt
»*(VJ - β* .
D D
Sat:
ω - w°V , (6.45)
а «Ж-valued one-form on U. Similarly, eet
Ω - 0*V , (6.46)
a JIT-valued two-form.
Theorem 6.6. ω and Q are globally defined» independently of the choice
of moving frame used to define them.
Du - Q (6.47)
where D is the differential operation on Lie algebra valued one-forme
defined above.
Proof. Mote that:
«(JT) 0
u(V) - V
for V € JT .
These two relations show that u is indeed defined independently of the
moving frame. (6.47) is now readily seen to be equivalent to relations (6.42)
(6.43).
We have just seen that there is an elegant "global" structure for K-
invariant connections on the principal bundle X. Let us now examine how
things look from tha point of view of the base space Z.
Let 0 be an open subset of Ζ and let
ν u *x

APPENDIX 1 - CONNECTION THEORY 449
be a cross-section map. Set:
ωυ " γ5ίωϊ (6.48)
% " γυίΩ> ' t6,49>
where ω and Ω are Λ1-valued differential forms on U.
Let U' be another open eubeet Ζ with a croee-eection γ , and
ω |fQ , defined similarly. Then, if U and U' are non-empty, there is a
map
ο ι unu· -* К (6.50)
such that:
γυ " <Wu· ■ t6*51)
Hhen the U,U' are chosen as open sets fron an appropriate open covering
of X, the system
defines a cocycle condition needed to define a principal bundle in the aenee
of Steenrod [19). It is readily seen that this bundle is isomorphic to the
one with which we began. The transformation law for the ω in terms of
the ω and α , are readily derived fron (6.47), (6.50), and (6.51).
It is non-tensorial, i.e., involves the derivatives of the Y.-.t ■
7. THE BIAMCHI IDENTITIES
Keep the notation of Section 6. Consider again the structure equations
(6.42)-(6.43)s
do) - ω α ω (7.1)
do»* - λ? ub α ω° + Ω* . (7.2)
Let us derive relatione by applying the exterior derivative to both sides of
(7.2):
0 - 2λ* dwb a uc + dfi*
be
" 2λ* (λ^ (/лы*^) A UC + dfi* r
be de
whence t

450
APPENDIX 1 -- CONNECTION THEORY
dfi* + 2λ* Ob a uC - 0 . (7.3)
be
Equations (7.3) are called tha Bianahi identities. (7.2)-(7.3) may be thought
of as a set of differential equations for (ω ,Q ). Very little is actually
known about the set of solutions, except in the simplest case, say
λ* - 0 . (7.4)
be
Equations (7.4) are equivalent to the condition that
[ΛΓ,ΛΓΪ - 0 , (7.5)
i.e., CK is an abelian Lie group. Equations (7.2)-(7.3) became decoupled
а а
dm - ίϊ
dfi
(7.6)
а
dFft
*(Ρ*Ϊ
- 0
- оа
л Л
in this case, the set of solutions is governed, locally, by the Poincari
lemma, and globally, by de Rham' s theorem.
We will now briefly review (without proofs) standard material in case
(7.5) is satisfied.
The curvature forms Q* satisfy (7.6) together with tha condition
JITJ Ω* - 0 , (7.7)
this implies that there are two-forms F on ζ such thatt
(7.8)
(7.9)
Conversely, given a set of two-forms
F**1 ,..., Fn (7.10)
on a manifold Z, satisfying
dFa - 0 , (7.11)
we ask: When are they the curvature forms on a K-invariant connection in a
principal bundle with structure group K? The answer (given first by A. Weil)
is well known, involving a bit of elementary fiber bundle topology, and has
played an impertant role for thirty years in algebraic gee·»try, quantum
mechanics ("geometric quantization"), and Lie group harmonic analysis (Kirillov-
Kostant-Auslander theory of unitary representations of nilpotent and solvable

APPENDIX 1 -- CONNECTION THEORY
4S1
Li· groups). It also turns out to be th· appropriate geometric formalise to
express Dirac's "quantization" condition for «agnatic monopoles.
Write the F's satisfying (7.6) vectorially:
i+l
(7.12)
·()
П— m
Regard F as a two-differential form with values in the vector space R
By the Poincar* lemma, Ζ can be covered with open subsets
in each of which there is a matrix of one-forms
such thet;
ΑΒυ - F . (7.13)
Whenever U and U' are two elements of the open covering, which
intersect in ϋΠυ', we have:
deu - p
- deu '
d(eu-eu.) - 0 . (7.14)
Now, it is possible to choose the covering iu} so that it has the
following property!
Whenever a finite number of open subsets of the covering ._ _.
intersect, its intersection is diffeomorphic to к .
(For example, one can choose the и to b« aonvax with respect to an affine
connection of Z, by a theorem of J.H.C. Whitehead.)
Since UПП' is R , we can apply the Poincarfi leeaa again, and obtain
a vector-valued function f,„., such thet:
UU
"ou- ■ θυ" %' · ί7·16ϊ
In the intersection of three open sets of the covering.
U, 0', tT,

4S2
APPENDIX 1 — CONNECTION THEORY
we havei
dfuu· + "uu'ir + dfiru
cuu· + fu'ir + Vu " COMtant' "У vou'ir € Rn"W * (7*17)
This assignment
(U,U\lT) - v^,^ (7.18)
defines a two-cochain in the Cech cohomology of Ζ (with coefficients in
n-m
R ) with respect to this covering. From (7.17), ve see thet this cocbain
is a cocycle. By a theorem of Weil (a variant of de Sham's), this determine·
tnt of
h2(z, Rn"") ,
the second cohomology group of Ζ with coefficients in R Since «3Γ
n-m
is isomorphic (as a vector space) to R ,
determineβ an element of
Η (Ζ,ΛΤ) , (7.19)
the vector apace of the second Cech (or de Rham, to which it is isomorphic)
cohomology group of the manifold Ζ with coefficients in the vector βραΰ€ Ж.
(I apologize to topologiets for the choice of notation. In topology Ζ usually
denotes the integers.)
Remark. For more detail about this sort of argument, see the excellent
treatment by V. Gulllemln, s. Sternberg and N. Nallach (15,41]. These
treatises also contain much mora detail about the relation to Lie group
harmonic analysis and quantum mechanics.
In each U,U' such that UП u" is non-empty, construct the map
♦ ,ϊ υηυ' ■* & (7.20)
as follows:
♦w(I> " '»■ vi + -"+Cv„-» (7·21>
tftenall that (v,....,V ) ia a fixed baaia of Jf. tha Lie algebra of the
ι n-m
connected abelian Lie group K.)

APPENDIX 1 -- CONNECTION THEORY
4S3
Sine· the Lie group К is abelian, the вар
expi JfC -*■ К
is onto And a local difisomorphism. For each (U,U') set
kwv - "»«♦«,·> · <7·">
The central question is nowι
Can the system of the ik ,} be chosen so that
UU'
whenever U,U',U" are three elements of the covering ._ ...
which intersects» the following condition is satisfied:
к - к к
OU' UU" U"U'
If (7.23) is satisfied, then there is a principal bundle with structure
group K, by a standard construction. It is then readily verified that the
ίθ } considered as a differential form with value in J£, defines a connection
for this bundle, for which F ,...,F are the curvature forms. The
analysis of the condition that (7.23) be satisfied is now straightforward, but
lengthy. I will only stats the result in the following elegant fovmi
Theorem 7.1. Let К be a connected abelian Lie group with Lie algebra ЛГ.
LEt v.(K) be its fundamental group. Regard *. (K) as a subgroup of the
translation group of JC (as the kernel of the exponential map ЛС -»■ K) .
Let F be a JIT-valued two-form on a manifold Z, such that
dF - 0 .
(Of course, DF also equals zero, since JC is an abelian Lie algebra.)
Then, via the de Rham-Cach-Mail isomorphism, F determines an element ch(F)
of tha cohomology group
Η2(Ζ,ΛΤ)
of Ζ with coefficients in the vector space «Ж*. Then, a necessary and
sufficient: condition that Ρ be the curvature tensor of an invariant
connection on a principal bundle X with structure group К is that ι
ch(F) € H2(Z,»1(K)) , (7.24)
where * (K) is regarded as a subgroup of the abelian translation group of
the vector space «Ж.
Thus, we see that there are two obvious sufficient conditions that every
closed Jf-valued two-form be a curvature tensor:

454
APPENDIX 1 -- CONNECTION THEORY
w1(K) - 0 (7.25)
I.e./ К is slowly connected, or
The second cohomology groups of Ζ with coefficients
('.26)
in the integers is zero.
Looked at from another point of view, if neither (7.25) nor (7.26) is
satisfied, the condition (7.24) will give us a primitive sort of quantisation
condition for the "gauge field" F.
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APPENDIX 1 - CONNECTION THEORY
455
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be a Fiber Bundle, Proc. Am. Math. Soc. lb 236 (1960).
19. R. Hermann, The Differential Geometry of Foliations, Ann. Hath. 72,
445-457 (1960).
20. R. Hermann, Equivalence of Submanifolds of Homogeneous Spaces, Math.
Ann. 158, 284-289 (1965).
21. R. Hermann, Cartan Connections and the Equivalence Problems for Geometric
structures, Contributions to Differential Equations 3_* 199-248 (1964).
22. R. Hermann, The Differential Geometry of Foliations, II, J. Hath, Meoh.
11. 303-316 (1962).
23. R. Hermann, Existence in the Large of ParalLelism Homomorphisms, Trans.
Am, Math. Soc. 161, 170-183 (1963).
24. R. Hermann, Geometry, Physio в and Systems, M. Dekker, New York, 1973.
25. R. Hermann, Gauge Fields and Cartan-Ehreemann Connections, Part A,
Interdisciplinary Mathematics, Vol. 10, Math Sci Press, Brookline, MA
1975.
26. R. Hermann (ed.), Ricci and Levi-Civita'в Tensor Analysis Paper,
translation, comments, and additional material by R. Hermann, Math
Sci Press, Brookline, MA, 1975.
27. R, Hermann (ed.), Sophue Lie's 1884 Differential Invariants Paper,
commenta and additional material by R. Hermann, Math Sci Press, 1975,
28. r, Hermann, Yang-Mills, Kalusa-Klein, and the Einstein Program,
Interdisciplinary Mathematics, Vol. 19, Math Sci Prsss, Brookline, MA
1978,
29. R. Hermann, Differential Geometry and the Calculus of Variations,
2nd edition. Interdisciplinary Mathematics, Vol. 17, Math Sci Press,
Brookline, MA, 1977.
30. R. Hermann, E. Cartan'β Geometric Theory of Partial Differential
Equations, Advances in Math, Ъ 265-317 (1965).
31. s, Kobayaski, Transformation Groups in Differential Geometry, springer-
Verlag, Berlin, 1972.
32. S. Kobayaski and K. Nomizu, Foundation of Differential Geometry,
Vols. I and II, Interscience, New York, 1963.
33. J. Milnor and S. Stasheff, Charaoteristia Classes, Princeton Univ. Press,
1965.
34. G. Reeb, Sur Certains в Proprietes Topologiques dee Varietes Peuilletees,
Herman, Paris, 1952.
35. H. slebodzinski. Exterior Forms and their Applications, polish scientific
Publishers, Warsaw, 1970.

456
APPENDIX 1 -- CONNECTION THEORY
36. И. Spivak, Caloulus on Manifolds, W.A. Benjamin, New York, 1965,
37. и. Spivak. Λ Соярглкаплгж Tntroduation to Differential Geometry,
Brandeis Univ., Waltham, Mass, 1970.
38. N. S tee η rod, The Topology of Fiber Bundles, Princeton Oniv. Press,
Princeton, N.J., 1951.
39. s. Sternberg, Leaturee on Differential Geometry, Prentice-Hall,
Englewood Cliffs, N.J., 1564.
40. A. Trautman, Fiber Bundles, Gange Fields, and Qraviation, General
Relativity and Graduation, Vol. I, A, Held (ed.), Plenum Pub. Co.
1980.
41. N. Hallach, Sympleatio Geometry and Fourier Analysis, Hath Sci Presa,
Brookline, MA, 1977.

APPENDIX 2
СЛЯГМГ 8 METHOD OF THE MOVING FRAME AS A GENERALIZATION
OF KLEIN'S "ERLANGER PROGRAM"
1. INTRODUCTION
Cartan developed the method of the moving frame ("repere mobile") for
two reasons: As ал alternative to classical tensor analysis, which avoids the
"debauch d' indices" that he criticises in the preface to this book, and to
generalise Klein's work [1,2], which deals (in modern language) with geometries
invariant under transitive Lie group actions on manifolds. While Cartan was
able to do brilliant computational feats with this formalism, it is also
difficult to understand, and has not survived in detail the transition since
1945 to coordinata-free, calculua-on-manifolds oriented methods of differential
geometry. In fact, it was Cartan'β own student, Ehresmann, who showed at the
end of Cartan'β life [3] how to work in this framework. The treatises by
Sternberg [4], Kobayashi and Nomizu [5], Hicks [6], Lichnerowiz [7], Bishop
and Crittenden [8], Bishop and Goldberg [9) and Kobayashi [10] now cover some
of this material from the fiber bundle and calculus-on-manifolds point of
view. The book by Guggenheimer [11] contains material that is very useful in
understanding the relations between Cartan, Lie and the classical differential
geometry of curves and surfaces. Schouten's treatise [12] contains an
exposition of Cartan's methods in the language of tensor analysis. The
lecture notes and papers of S.S. Chern [13] were very influential in making
Cartan's methods accessible to the post-1950 mathematical generation. In
this essay I will try to explain, in modern terms, some of the mathematical
background to Cartan's work.
Let X be an (η-dimensional, С , paracompact) manifold. Choose the
following range of indices with summation convention:
1 <_i,j <n .
We will use the version of the summation convention that tensor-analysts like,
using "upstairs" and "downstairs" indices with repeated pairs of indices
summed if one element of the pair is upstairds, one downstairs. Thus
* n t
» - Σ * ·
i-1 λ
The symbol Α.. means the value of the symbol (Aw) *t i " j# and is not
summed.
iir..i.
AJJl'"Jp
457

458
APPENDIX 2 -- METHOD OF THE HOVING FRWC
is the indexed symbol
which is defined as follows:
1ι···1Ρ Д ϋι·"1ρ
Definition. A coordinate system for X is an (n+1) -tuple
(Y: x1 χ") Ξ (и, x1)
such that:
a) Y is ar, open subset on X
1 »
b) Each χ is a real-valued, С function on Y.
c) The map
у ■* (χ (у) χ (у))
of Υ ■* R is a diffeomorphisra between У and an open eubaet
Let (x ) be such a coordinate system. It will be understood that
everything takeβ place in an open subset У of Χ* The reader who wants to
see everything stated precisely will have to keep tracX of these matters for
himself. Geometry is the poetry of mathematics and geometers claim poetic
licence about such details I
The one-differential forms
a*1
can then be constructed. Our assumption that у -*■ (χ (у),...,х (у)) is a
diffeomorphisDi of Υ with an open subset of R implies that, at each point
у € Υ, the one-covectors dx (y),,,,,dx"(y) form a basis for the cotangent
space
xd
У
d
to X at у. (X is the dual vector space to the tangent space X to X
У L У
at y.) Thus, there is a basis for X which is dual to the dx , We label
this basis as follows:
3l(y) .

APPENDIX 2 — METHOD OF THE MOVING FRAME
459
As у varies within Y, we obtain η cross-sect ions
Э1 Эп
of the tangent vector bundle T(X), which satisfy the relations:
<dxi, Э > - 6L . (1.1)
By a principle of "calculus on manifolds", cross-sections of the tangent
bundle T(X) above open subsets of X can be identified with derivations
V(Y) of the algebra ДОУ) of C*\ real-valued functions on Y. The
following relation links* by duality, tangent and cotangent bundles and the
identification of the cross-sections of the tangent bundle with derivations
of #-<Y)i
<dfi V> - V(f) . (1.2)
He also write v(f) as j? (f), read the Lie derivative of f by v.
Thus, we can rewrite (1.1) as follows:
V " V
<dxi, Эл> - ЭЛх1)
- 6L
(1.3)
When we want to emphasise thet the vector fields Э. depend on the coordinate
i i
system (x ), we also use the Leibnizian notation
t? г ^ ■
In modern differential geometry, tensor fields are cross-sectiona of
the tensor products
T(X)®---®Td(X)®···
of the tangent and cotangent bundles. (See the treatise by Steenrod [15],
Kobayashi and Nomizu [5], and Milnor-Stasheff [16] for the language of fiber
bundle theory that we use.) The tensor products
τ?Γ"?ρ " 3i ®'"®Э. ®dxJ1e-"®dxjP
then form bases of the tensor fields, restricted to the open subset Y.
Algebraically, they are bases of the ДОУ)-modules obtained by tensoring
the free modules V(Y) and & (Y) a certain number of times. Tensor
analysis, as codified for example in Schouten's treatise [12], deals with
tensor fields in terms of such bases.

460
APPENDIX 2 -- METHOD OF THE MOVING FRAME
The historical source for such of this is * paper written Is 1900 by Ricci
and Levi-Civita [17], which I have translated and combined with some of my own
exposition. Levi-Civita's book [19) is also stLll valuable. It is, in fact
much «ore accessible to the contemporary reader than Schouten's treatise
and has the great mathematical advantage of being linked to physics. I
suspect that tensor analysis developed ita nore baroque qualities whan the
second generation lost contact with the physics. To this day, physicists
have a soft spot in their heart for tensor analysis, and the expositions of it
in relativity books are oftan quite readable and intuitive.
Cartan's idea can now be described sinply, but imprecisely. Replace
i d
the local cross-sections (dx ) of the cotangent bundle Τ (X) by other
cross-sections
(ω1) .
which are independent at each point, but which are adapted to the geometric
situation.
Thus, in Riemannian geometry, the natural choice is that in which the
metric tensor takes ita algebraLc canonical form. Classical tensor analysis
would wxita a Riomannian matrie ast
de2 - g dx1 «dx3 (1.5)
9ij " 9Ji "
The coafficienta (g. ) are functions on Y, which define a two-field,
sysntric, covariant tensor field. Let us for simplicity restrict attention
to positive metrics.
2
in fiber bundle tanas, Mds " swans a symmetric, positive-definite
bilinear form
(νχ.ν2> - <vx.v2>
in the tangent bundle to X:
<VV - gij(y)<dxi(y), v1><dxj(y), v2> (1.6)
for all у € Υ, ЧуЧ2 € Υ .
The componenta Ϊ9ι^) of the metric are then functions in the local
neighborhood Y. To study how the metric (1.6) varies as у varies, one ssist
now differentiate these functions in the local coordinate system (x ). It
is this that creates the massiness of tensor analysis, the "debauch d'indices".
Cartan proceeds as follow* (in aodsrn language)ι

APPENDIX 2 — METHOD OF THE MOVING FRAHE 461
Definition. Let Td(X) ·ΤΛ(Χ> be the fiber bundle over X, whose fiber above
d
a point χ of X is the symmetric tensor product of two copies of the space X of
one-tangent covectors. λ (positive) Riemannian metric is a (smooth) cross-
section Nds " of this bundle. It associates to each point χ of X a
symmetric, positive-definite inner product
(v ,v ) ■* ω(ν ,ν Ϊ (1.6)
on tangent vectors.
An (orthonormal) moving frame for such a metric is an (n+1)-tuple
(Yf ω1 ω") = (Υι ω1) (1.7)
consisting of an open subset Υ of X, and a set
1 η
ω ,...,ω
of one-forms in Y, which are linearly independent at each point of Υ and
can provide a "canonical form" for the metric (1.6), in the sense that!
η
Σ
i-1
η . .
«VV " Σ ω (ν1ϊω1(ν2ϊ (1.8)
for all ν,,ν € Χ ι у € Υ .
Remark. A useful alternate notation for (1.8), using the summation convention,
would bet
ds2 - β ω1 . ω3 , (1.9)
where · denotes the symmetric tensor product of one-forme.
Carten now proceeded to study the "differential" part of the metric
geometry by investigating the relations between the one-forme ω and their
exterior derivatives dui , independent of local ooordinatee. In fact, for
Riemannian geometry these relations are remarkably simple.
Main Theorem of Riemannian Geometry. For each orthonormal frame (Yi u ) of
the Riemannian metric, there are uniquely aseociated a set (uri Or) of
differential forme in Υ such that:
du. ■ ω. л ω.
л J * J «J
atu ■ ω л ы + Qr
i i к i
(1.10a)

462 APPENDIX 2 « METHOD OF THE MOVING FRA№
uj + ω. - О
fl? * О1
(1.10b)
■i "j
These are called the Structure Equations of Riemamian Geometry.
what fiber bundle theory added to this was a way of constructing a
manifold В of dimension n(n+l)/2 (associated with the metric) a fiber
space шар
us В ■* X ,
and a set (θ., θ?, Β3.) of differential forms on В satisfying the following
conditions:
a) The (0{'0f) are one-form·, and are linearly independent at each
point of B.
b) dBL - θ^ л ω
Ле{ - β^Αβ****
.?4β1
θ| + Qj - 0
с) То each orthonormal frame (U: ω ) of the metric, there is a
(1.11)
aroas-eection map
γ: U -► В
such that:
γ'ίθ1)
Τ*(θ|ϊ
Τ*(κ|)
i
■ ω
■ -г
(1.12)
The important point for the geometer is that the manifold В and the
forms (β ,ΘΤ,θ:) it carries, is intrinsically associated with the metric
2 i i i
ds . Roughly, this means that to each diffeomorphiam φι Χ ■* X' that carries
one Riamannian mptric ηηϊ-o *nnthwr, th»re is an associated diffeoaorphiam
φ· ι в -*■ В' between the associated bundles, which carries the set of form·
defined in В into that defined on B'.

APPENDIX 2 - METHOD OF THE MOVING FRAME
463
Notice also why Cartan liked to use forme rather than vector fields,
i.e., "covariant" rather than "contravariant" tensors. The pull-back mapping
γ* is required, and it is not defined for vector fields.
I cannot hope to give here a complete exposition of Cartan'a work from
the modern fiber bundle, calculus-on-manifolds point of view. My plan is to
supplement what Cartan himself presents in this volume with certain material
that I have found useful in understanding his work, since "moving frames"
are sets of one-forms that form a basis of forms at each point, and the
proper name for the geometric structure defined by such bases is absolute
parallelisms, I will concentrate on their study and the role they play in
Cartan*s work on geometric structures.
λ special case of spaces with such moving frames, which are globally
defined, are the Lie groups. The moving frames—here called Maurer-Cartan
forme—are bases of one-forms that are invariant under right or left
translation on the Lie groups. As the name indicates, Cartan encountered these objects
early in his career—his famous thesis was on Lie groups—and much of his
geometric work can be thought of as an attempt to define "geometries" by means
of algebraic relations between sets of one-forms and their exterior derivatives,
as the Lie group structure is defined by the Maurer-Cartan equations,
relating the Maurer-Cartan forms and their exterior derivatives. In this way,
Cartan hoped to extend the ideas Felix Klein expressed in his "Erlangen
Program" Lecture [1] about geometries (e.g., Euclidean, projective, conformal,
etc.) defined on homogeneous sp ces of Lie groups to spaces that "approximately"
had such a structure.
Remark to Physicists, what Cartan calls a "moving frame", physicists often
call "quasi-coordinates". Cartan'a ideas on generalizing Klein*s work is
often referred to as a "non-holomonic" version. I have never been able to
understand what this term means I
2. AFFINE CONNECTIONS IN THE SENSE OP KOSZUL
To begin the formal study of "absolute parallelisms'' I will begin by
briefly recalling the theory of affine connections on manifolds in the quasi-
algebraic form daviesd by J.L. Koerul [19]. This formalism is extremely useful
in differential geometry, particularly in Riemannian geometry and Lie group
theory, and has been developed in many places since Koszul's work.
Let X be а С , paracompact, η-dimensional manifold. Let ДОХ)
m
denote the С , real-valued functions on X, considered as a commutative
associative algebra (under point-wise multiplication) over the real numbers.
The vector fields V(X) are then the derivatives of the algebra ДОХ). The
operation which lets a V С V(X) act on an f € ДОХ) is often called Lie
derivative and denoted by Д* (f).

464
APPENDIX 2 - METHOD OF THE MOVING FRME
Consider jT(X) as the zsro-th degree differential forma. Let 0r(X)
denote those of degree r - 0,1,2,... Lie derivation by a V С ψ {Υ.) can
then be extended to 0r(X)-.
3?yi »r(X) - »Г(Х)
satiefying the following conditions
^ν(ω1Λω2) - 3?v^v) л ω2 + ωχ л &уЫ2)
&уШ) - d*v(«)
for ω,ω ,ш С £Р(Х). The Jaadbi braaket (aleo called "Lie bracket") between
vector fields
tVl'V "* IV1'V
(which raakee T^(X) into a Lie algebra) can aleo be considered as a Lie
derivative
^νίνι} ■ Iv,vi3
for νχ € ТЧХ) .
The following relation linke Lie derivative on vector fields and their
duals, for forma:
ДГ <<ufV >> - <i?v(U), νχ> + <ω, [V,V1)>
The Lie derivative is a "natural" differential operation, in the
preciee eenee of category theory. (Intuitively, it ie "independent of
coordinates".) Thie ultimately derLvee from its geneeie in transformation
group theory, а» a "dragging along" relation. ("Liouville's theorem" in
claseical analytical mechanice ie the prototype.)
In local coordinatee (x ) for X, Lie derivation of vector fielde
taJcee the following form;
V ■ Ai3i (2.1)
νχ - Bi3i (2.2)
[V,V13 - (Ai3i(B Ϊ -Bi3i(A ΪΪ3 (2.3)
Notice that the componente of the vector fields are both differentiated in a
Hak*v-ayn**tric" way. It ia thia that ^aaruttMi tha "ttnsorlttl" natura of
the operation.

APPENDIX 2 - METHOD OF THE MOVING FRAME
465
However, there is often the need for another sort of differentiation of
one vector field by another in which the components of one of the vector fields
are not differentiated. Koszul [19) set it up in the following way:
Definition. An affine connection for the manifold χ is an B-bilinear map
Г(Х) * Г(Х) -► Г(Х)
(V,Vl) - 7ννχ ,
satisfying the following conditions:
7fvvi " CTvvi 12·4>
for ν,νχ € Г(Х) , f € βΤ(Χ>
W ' ^νίί>νι + f7v νι t2*5>
for ν,νχ € Г(Х) , f € ДОХ)
using these rules and the local representations (2.1) and (2.2) for
V and V., we then have:
7vvi ■ A\ tBJV
(2.6)
- Ai3i(Bj)3 + aVv3 ο ϊ .
In these local coordinates the functions Γ such that
ν„ О.) - Гк Э. (2.7)
3i j ij k
then determine the operation 7. Substitute (2.7) into (2.6) to obtain the
definitive form of the operation in local coordinates:
7V Vl " <Ai3l(Bk) +AiB3r|[j)3Je . (2.8)
Remark. Relation (2.8) is the starting point for the classical tensor analysis
[12,17). One starte by "postulating" a collection of such Γ'β in each
coordinate system, and defines "covariant differentiation" to essentially
agree with (2.8). (That is what "V" i« supposed to indicate.) One must then
postulate a transformation law for the Г under change of local coordinates
so that (2.8) remains "invariant". This requires a "non-tensorial" change of
the Г, i.e., they may vanish at one point in one coordinate system, but not

466
APPENDIX 2 - METHOD OF THE MOVING FRAME
in another. Koszul essentially algebracized these ideas with an enormous
bonus of mathematical and conceptual clarity.
A basic point of classical tensor analysis is that "covariant
differentiation" differs algebraically from "ordinary differentiation" in that the
result of two differentiations is not independent of the order, unless the
connection is "flat", i.e., equivalent to the "standard" one on Rn. We will
summarize what is known about this {without proof) in the following way ι
Theorem 2.1. Let 7ί 1C(X) χ ΉΧ) ■+ Τ'ίΧ) be an affine connection on a
manifold X. For vlfV € T^X) set:
«<vv ■ 4 4 - Vvr 7'vv u'9>
H(V ,V ) i« then «n ДОх^Нпааг map of ■V(x) ■* ^(X) . «i« map
tVl'W "" RiV1'V2}(V3>
is an ^(X)-trilinear map of the #"(X)-module V(X) into T'U). R is
called the curvature tensor of the affine connection 7. (Its "tensorial"
nature la equivalent algebraically to ita linearity with reapect to the
module structure on ψ"{ Χ).)
There is another tensor field one may associate with 7, called the
torsion and denoted by T:
TtW ' 7v V2 " 7v Vi " IV1'V " 12.10)
Τ is an ^(X)-bilinear map of T'(X> x УЧХ) ■* УЦХ).
One might ask if there are any more tensor fields one may associate
by such differentiations and algebraic operations with affine connections.
There are; The covariant derivatives of R and T, which we will not
describe in detail here. The following remark (which we shall not prove
here either) is a sort of answer to thist
Theorem 2.2. Let (X,7), (X',7') be affine connections on the real-analytic
manifolds X and X*. Suppose that
a) V and 7' are real anal/Lie
b) There are points χ € Χ, χ' € X' with a linear isomorphism
between tangent spaces X and X',, which carries the
corresponding values at χ and x' of the curvature and torsion tensors
and their covariant derivatives into each other.

APPENDIX 2 - METHOD OF THE MOVING FRAME
467
Then there are open neighborhoods Υ of χ in X, Y' of x' in X',
and a real-analytic diffeomorphism
♦ : Υ -► Υ'
that carries one affine connection into another. If, in addition, the
connections are aorrplete in an'appropriate sense, and if X,X' are simply connected,
then Υ can be taken as X, Y' as X'/ i.e., the connections are globally
equivalent.
3. THE CURVATURE TENSOR AS THE INTEGRABILITY CONDITIONS FOR PLAT MOVING
FRAMES OF AFFINE CONNECTIONS
Continue with the notation of Section 2. Let V be an affine connection
Cor the manifuLd X. For V С V(X), the coverLent derivetive uperetion сел
be extended to the differential forms:
(VV«KV1(.'..,V) - ^ν(ω(νι Vr>) - u)(Vv νχ Vr>
(3.1)
_ ... _ U(V1#...,Vv Vr)
for ν,νχ V € V(X) , ω £ »Г(Х) .
A differential form ω (or general tensor field . for that matter) is said
to be flat if the following condition is satisfied:
Vyfc) - 0 (3.2)
for V € ТЧХ)
Let us consider the differential equation for flat one-forms in a local
coordinate system for X, labelled (Y,x ). If
χ »j ■ ΓΪΑ · (3·3>
then
<v3 (dxj), ak> - j?3 (<dxj,ak>) - <dxj,v3 ak>
- °- <<5χ3'ΓίΛ>
--* -
Hence,
Vi(dxj) - - Г^ dxk (3.4)

468 APPENDIX 2 - METHOD OF THE MOVING FRAME
ω - a dx
V. (ω) - Э ia.) dxj + лл V. (dxj)
3i j i j 3i
- Э (a ) dx^ - а Г^ dx
V i' j iX
Hence, the differential equations for flatness of V are:
Theorem 3.1. If the curvature tensor R is identically zero, than the linear
partial differential equations (3.5) for the function (a ) are completely
lntegrable In the classical sense. At each point χ € X, there is an open
neighborhood Υ of χ in X, and a unique solution (a ) of (3.5) in Τ
such that (a, (x)) takes prescribed values.
Proof. Since
Oif3.] - О (Э.6)
wa have ι
Va V - V 7 (3.7)
\ 3j 3j 3i
sett
ω -► V. (ω) - D. (u)
3i i
D. is a linear, first order differential operator on the set of 'a. (3.7)
says that
[Di#D ] - 0 (3.8)
Equations (3.8) do indeed express the "inteqrability", in the classical sense,
of the first order, linear partial differential equations (3.5), which take
the form
Di(ui) - 0 .
Q.E.D.
Remark, ibis result can be obtained from the Frobenius Complete Inteyrability
Theorem [14] using connection theory in fiber spaces. A flat one-form is a
cross-section map

APPENDIX 2 -- METHOD OF THE MOVING FRAME 469
x -► Td(x) ,
whose graph «ay be obtained as a leaf of a foliation on
X*Td(X) .
There is also a proof available using the concepts of parallel transport
associated with an affine connection, and the relation between curvature and
parallel transport around a closed rectangle. №« following result can then
be obtained t
Theorem 3.2. Let 7 be an affine connection, on a simply connected manifold
X, whose curvature tensor is «его. Then, there are one-forms (u ) globally
defined on X that are flat» with respect to the affine connection 7, and
that are linearly independent at each point of X. Thus, the
form a besis for the ^(X)-module »1(X) .
Here is the converse:
Theorem 3.3. Let (ω ), 1 <_ i,j,k <_ n, be a set of one-forms on the n-
dimsnsional manifold X that are independent at each point of X. (Thus,
the (ω ) form a moving frame in Cartan's sense.) Then, there is a unique,
curvature zero affine connection 7 on X such that
7у(«Ь - 0 (3.9)
for all ν € T^X) ,
i.e., the (ω ) are flat with respect to the affine connection.
Proof. Let (V ) be the vector fields that are dual to the u , i.e.,
<ω\ν > - β* . (3.10)
It is readily proved that the (V ) form a basis for the ^(X)-module V(X).
«hue, a pair <V,V) of elements of V(X) can be written in the form:
V - A V.
V ■ BVi
Set
7 V - A1^ (Bj)V, . (3.11)
ν νΑ 3

470 APPENDIX 2 — METHOD OF THE MOVING FRAME
It is readily verified that 7 is an affine connection for X. For V - V ,
V - V , (3.11) gives
?v (V) - 0 . (3.12)
i
The curvature may now be determined. Suppose:
[VV.J - ^Д . (3.13)
with
Then,
fjj С *ЧХ) ■
R<VV<V " \{y\» · 7vjt\(V> " 7[vi#Vji (V
- 0 - 0 - V (V)
f V
ij 1
■ - fij \<v
(3.14)
Since B( , )( ) is ^"(X)-trilinear, to prove it vanishes identically, it
suffices to prove that it vanishes in a module basis for V(X). Equation (3.14)
does precisely this.
The proof of uniqueness of 7 is left as an exercise. Q.E.D.
Let us compute the torsion of the affine connection defined by (3.9):
TtW ■ \ v3 - 'Vj vi - (VV
- 0 - 0 - f*. V,_
Id *
■ - f ν
rij vk
Now,

APPENDIX 2 - ICTHOD OF THE MOVING FRAME
471
Thus,
ω1(Τ(ν ,VΪΪ - - du^V ,V) (3.15)
Since both sides of Equation (3.15) depend #"(X)-bilinearly on V ,v. »
(3.15) holds for arbitrary V ,VR € ТЧХ).
Now, Τ is а V(X)-valued two-form on X. Hence, the inner product
with ω is a two-form that we denote as
<ιΛτ>
tfe have then proved the following result:
Theorem 3.4. Let (ω ) be a moving frame for the manifold X, defined as a
est of one-form on X whoee value a at each point χ of X form a battle of
X . Let Τ be the torsion tensor of the affine connection 7 on X such
x i
that the ω have zero covariant derivative. Then, the following relation
links Τ with the exterior derivative of the ω :
<ωλ,Τ> - - du1 . (3.16)
Me can now investigate the conditions that the covariant derivative of
the torsion tensor be zero
V (Τϊίν.,ν Ϊ - V (T(V ,V )) - T(VVV.,V ϊ - T(v , Vv (V ΪΪ
V, J К V, J * i J * ^ J
- \«l>\ ■
Me have then proved;
Theorem 3.5. The covariant derivative of the torsion tensor of the affine
connection defined in Theorem 3.4 is zero if and only if the functions f .
i i ^
defining the exterior derivative of the ω are zero. The (ω ) then
satisfy the Maurer-Cartan equations of Lie group theory: The (f .) are the
structure constants of a Lie algebra.
In summary, in this section we have developed certain relations (due
essentially to Schouten and Cartan [20], but presented in modern farm by
Koszul [19] and Nomizu [21]) between absolute parallelisms, curvature-zero

472
APPENDIX 2 - METHOD OF THE MOVING FRAME
affine connections, and Li· group theory. This material provides one
foundation for our treatment of the Method of the Moving Frame.
4. LOCAL MOVING FRAMES AND THEIR TRANSITION GROUPS FOR COSET SPACES OP LIB
GROUPS
Let G be a connected Lie group, Η a closed Lie subgroup. Let Η act
on G by multiplication on the right:
(g,h) -*· gh*1 (4.1)
for g € G, h € Η .
This action is free, and the orbit space X is called the aoaet враое G/H.
See the treatise by Helgaaon [22], as well as tha differential-geometric
treatises cited above, for the differential-geometric theory of coset spaces.
The orbits are subsets of G of the form:
gH .
Let w: G ■♦ X be the map which assigns to g € G the orbit gH to which it
belongs. It is known that X can be given an analytic, paracompact manifold
structure, so that w is a eubnereion, e.g., wA(T(G)) - T(x). ν is also a
principal fiber bundle, with Η as structure group.
By tha implicit function theorem, at every point χ С X there are
open neighborhoods Υ containing x, and cross-section maps γ: Υ -*■ G. We
obtain bieaa of the» differential one—fо me on Y, i.e., moving frosMS, by
pulling back the Cartan-Maurer forma on G. If γ': Y' -*> G is another cross-
section over another open subset Υ of X such that YflY' is now empty,
we plan to determine how tha moving frames in YftY" obtained by pulling back
the Maurer-Cartan via γ and γ' are related.
Now, for χ € YflY', γ(χ) and γ'(χ) lie in the same fiber of w,
since they are both cross-sections of the map w. Hence, there is an element
h(x) С Η such that:
Y*(x) - Yix)h(x)"1 . (4.1)
As χ varies in ΥΠγ·, we obtain a map hi YflY" η, such that
h(x) - h(x) .
Choose indices as follows, and the summation convention on these indices:
1 <_ i,b <_ m - dim G
1 <_ i,j ^ η - die χ

APPENDIX 2 - METHOD OF THE MOVING FRAME
473
m+ 1 <_ ufv <_ m
m- η - dim Η
Let
(ω*ϊ , (ω*ϊ
L· η
be bases of the left and right invariant Maurer-Cartan forms on G. Thus:
.e . e b с .. _.
aar - λ. ωτ Λ ω (4.2)
L DC L
. a .a b с ,.
% ■ XbCURAWR (4-3>
with (λ. ) the structure constants of the Lie group G.
be
For g € G, let
V9"
ν
;ft and
LgtU?
V^
LgltaU
у$
+ 99'
, -1
-* g'g
right ectlon o;
a
■ UL
a
■ "R
" <***>ί«ΐ
" ««>£"£
(4.4)
(4.5)
(4.6)
(4.7)
where (Ad g). is the matrix of the adjoint representation of the Lie group
b
G on the Lie algebra 9. (The Maurer-Cartan fonts may be considered as
elements of the dual vector space to 9.)
Example: G - GL(tf£, R), the grcup of real nonsingular matrices.
1 < α,β < /υ
4
%
UL
-
-
-
s>
. -1
dgg
-1.
g dg
(The components of the matrices af one-forms are the actual Maurer-Cartan
forms.)

474 APPENDIX 2 -- METHOD OF THE MOVING FRAME
L*^) - (g^)"1 d(g0g)
-1 .
- g g0<*g
-ι .
- g eg
в ω
L
g0 R R
L* (« ) - d(g g)(g g)
gQ R oo
-1
- gQdgg gQ (4.8)
- gQg dggo* t4-9>
- Ad дЫь)
Relations (4.8) and (4.9) prove, in this exanple (which is, in fact, typical)
л л
that the (u ) and (ω ) transform under the same linear representation—
L R
in fact, the adjoint representation—under R and L_, respectively.
G G
Return to the case of a general Lie group G and ooset space G/B - X.
The left action
(g.g0) - gg0
of G itself passes to the quotient to define the transitive action of G on
G/H - X. In other words, the projection nap w: G -»· X intertwines the left
action of G on itself and the (only) natural action of G on X.
Theore» 4.1. There is a natural action of the product group G* Η on the
space of local cross-section naps γ: Υ -*> G. Hanely, if g € G, h € H, than
(g,h)(Y)(x) - gig'^h . (4.10)
Proof. He must show that formula (4.10) defines (g,h)(γ) as another
cross-section nap. Apply the projection nap w: G ■♦ X to both sides of (4.10) ι
n(g,h)(Y)(x) - wtgYtg'^h)
■ *(g>(g χ))

APPENDIX 2 -- METHOD OF THE HOVING FRAME
475
(since π is invariant under right translation by H)
- w(y(x))
(since w intertwines the left action of G on G
and the action of G on G/H - X)
- x #
since γ is a cross-section of w.
Q.E.D.
We can also consider these local cross-sections from the point of view of
foliation theory.
Theorem 4.2. The fibers of w are the leaves of the completely integrable
Pfaffian system ( = foliation)
i
ω. - 0 .
L
Proof. The fibers of w are (by general principles) the orbits of the
Lie algebra Jr of vector fieIds, i.e., the vector fields which are invariant
under left translation and whose orbit curves are of the form
g -*> g exp(th)
h € ЛГ.
The (ω ) are just the dual Pfaffian forms. Q.E.D.
L
Theorem 4.3. If γ: Υ -*> G, ycx, is a local cross-section of the submersion
map wi G ■♦ X, then the one-forms
γ*(ω*) (4.11)
L
form a basis for forms on Y, i.e., a moving frame for Y.
Proof. This follows from general geometric principles. γ(Υ) is a
submanifold of G which is transversal to the fibers of w, hence the
independent forms on the left hand side of (4.11) (whose vanishing defines
the tangent subspace to the fibers of w must be independent on γ(Υ)).
Let us now calculate how the moving frames
and
Y"h£)

476 APPENDIX 2 - METHOD OF THE HOVIHG FRAME
differ, where
Yf Υ ■* G
γ": Υ' ■♦ G
are two cross-sections, where Y,Y' are open subset* of X such that:
γΠγ· is non-empty .
Thus» we have;
Y'(x) - γ(χ)Μχ)"1 (4.12)
where h: UflU' Η is a map. (h is what the physicists call a gauge
transformation.)
To calculate
γ·*(ω*> , γ'* (ω*)
L R
we can use the ■symbolic" representation for the Maurer-Cartan form:
ы » g dg (4.13)
γ'·(ω ) - (Y(x)h(x)-1)"1 diyixihix)"1)
- h(x)Y(x)(d(Y)h"1-Yh'1dhh"1)
- Ad Μχ)(γ*(ωΓ)) - dhh"1 (4.14)
We can now translate this into a general result.
Theorem 4.2. Let γ,γ* ι Υ -*> G be cross-section паре on an open subset Υ of
X - G/H, and let
h: Υ ·*· Η
be а шар from Υ to Η such that:
γ'ίχ) - Yix)h(x)"1 (4.15)
for χ € Υ .
a a
Let (ω), (ы ) be the left and right invariant Maurer-Cartan forms of G.
L R
Then,
γ·*(ω?) - Ad(h(x))* Y*(ui) - h*(u>*) C4.16)
L D L· R
for 1 < a,b < a ■ dim G .

APPENDIX 2 - METHOD OF THE MOVING FRAME
477
In particular, for
a - i ,
1 <_ i < η ,
where ω are th· Maurer-Cartan forme which annihilate H, we have:
Y'Mub - Ad(h(x))* γ*(ω?) + AdihU))1 γΜω") (4.17)
L· j L· U L
5. THE MAURER-CARTAN FORMS AND STRUCTURE EQUATIONS FOR AFFINE SPACE IN TERMS
OF FRAMES. AFFINE CONNECTIONS IN CARTAN'S SENSE.
Cartan did not use the modern, abstract defintion of a Lie group to
calculate the Maurer-Cartan forme. He described the groups more concretely in
tame of "frames". I hope to show that this is still a very efficient and
intuitive method for calculating! Guggenheiner's treatise [11] also contains
extensive material along these lines.
Let X be an n-dimanaional real vector space. Choose indices as usual:
1 <_ i, j <_ n. Let L(X,X) be the vector space of linear тара: X ·*■ X. Let
GL(X) be the group of such вере which are invertible.
Definition. An affine transformation of X is a diffeomorphism
+ : X ■*· X
of the form:
♦ (x) - Ax + xQ (5.1)
with:
А € GL(X) , χ £ X .
The maps of the form (5.1) form a group under composition, called the affine
2
group. It is a Lie group (of dimension η +n - n(n+l)) which acts transitivel
on X. Me denote it as:
AFF(X) .
The isotropy subgroup at the point χ - 0 is obviously the set of
transformations of the form (5.1), with χ - 0. Hence,
X - AFF(X)/GL(X) (5.2)
Identifying in this way GL(X) with a subgroup of AFF(X) enables us
aleo to describe the algebraic structure of AFF(X) as a group. Let Τ be
the subgroup of transformations of the form (5.1) which are translations, i.e.
such that A - 0.

478 APPENDIX 2 - METHOD OF THE HOVIMG FRAME
Theorem 5.1. Τ is an invariant subgroup of AF7(X).
Proof. Let Φ be of the form (5.1), and let τ,.τ' С Τ be of the follow-i
ing form: For all x X*
t(x! - χ + x. ,
t" (x) - x + x' ,
χ ■ ,x € X .
Then
♦ (T[x)) - A(x + x ) + xQ
- Ax + Ax + xQ
t'(#[x)) - (Ax+x0) + xj
Thus, ift
χ'χ - Αχχ , (5.3)
we have:
♦T - T'# (5.4)
i.e.,
♦V € Τ
which proves (since φ is an arbitrary element of AFF(X)) that Τ is indeed
an invariant subgroup of AFF(X).
Theorem 5.2. AFF(X) is isomorphic to the semi direct product of the abelian
invariant subgroup Τ and the subgroup GL(X).
Proof. Assign to φ of the form (5.1) the pair
(A, ) С GL(X) »T
witht
τ (χ) - χ + χ
Then, if ♦' - (Α* ,τ·)
♦ '(♦(x)) - ♦'<Αχ+*0>
- A' (AX+ JL·) + Χ*

APPENDIX 2 - METHOD OF THE MOVING FRAME
479
- A'A(x) + A'x + X*
for all χ С X.
But, using (5.3)
(ττ,)Α'1(χ) - x + A· (xQ) + x^
Hence, we have ι
♦ '♦ - (A'A, А' (ТТ')А·"1) (5.5)
which is the "semi-direct product" rule.
In order to calculate the Cartan-Maurar form for G * AFF(X), Cartan
constructed a space (called the affine frame bundle and denoted as AFft(X))
X on which G acts simply and transitively), so that G can be identified,
as a manifold, with afr(x).
Definition. An affine frame for the n-dimansional vector space X is an
(n+1)-tuple
f - (χϊ x. x ) (5.6)
ι η
where ι
χ С X
(χ,,... ,χ ) is a basis for X .
ι η
Let the affine transformation (5.6) act on AFR(X) as follows:
+f - (φχι Axlt...,Axn) (5.7)
- (Ax-fX-i Ax, ...,Ax )
0 η
Theorem 5.2. AFR(X) is a manifold of dimension
2
η + η - n(n+ 1)
Formula (5.7) defines a transformation group action of G on AF(X). ТЫ*
action is simple (i.e., if *f - f for f € AFR(X) , then φ - identity) end
transitive.
Proof. First, we must verify that (5.7) really does define a
transformation group action. Let φ € G, i.e.,
♦' (x) - A'x + x£
Then,

480
APPENDIX 2 - METHOD OF THE MOVIN6 FRAME
♦ '♦ - (A'A, Afx0 + x^) (5.8)
♦ '(♦(f)) - ♦'(Ax + xi Ax,,...,Ax )
(A* (Ax+xn) + x't ΑΆχ.,.-.,Α'Αχ >
, using (5.8)
(♦'•)(f) ,
which indeed shows that the formula (5.7) defines a transformation group action.
Let us prove simplicity. Suppose for some f - (xi x_,...,x )
(λχ+χ ι Ax, ,...,Ax ) - (x; x,,...,x )
hence:
Ax * χ - χ (5.9)
**! ■ Xl
(5.10)
Ax ■ χ
η η
Now, we have chosen (x, ,...,x ) so that they form a basis for X. Hence!
ι η
conditions (5.10) implies that
л - identity
(5.9) now implies that
χ ■ 0 ,
0
which shows that:
Φ - identity ,
i.e., G acts effectively.
To prove transitivity, note that the point χ - 0 can be transformed
under G to an arbitrary χ С X by the translation τ - (0,x). Q.B.D.
Define maps
Pi e, e : AFR(X) ■*· X
ι η
as follows:
P(xt x.,...,x ) - χ
ι η

APPENDIX 2 - METHOD OF THE MOVING FRAME 481
Vх' xl V ш *l
(S.lll
e (χι x.m.'iX) ■ x
π 1 π π
we use the differential-forms-with-values in a vector space formalism of
Appendix 1 to compute the exterior derivatives of the vector valued functions
(i.e., zero-th degree differential forms) defined by formula (5.11). Recall
that
«r(APR(X)i X)
denote the r-th degree differential form, with values on the vector space X.
It is
a>r(AFR(X)) 0 X .
Exterior multiplication of soatar valued forms defines a map
ι er(APR(X)) * »e(AFR(X)i X)
♦ »r*e(AFR<X)i X)
-*- (U Α θ) β Χ
u л (θ®χ) - (мл θ) β χ
Now,
Ρ € a°(AFR(X)i Χ)
Hence, there are n scalar one-forms
ω1 € ^(AFRiX))
such that:
dP - wiAei - (5.12)
Similarly, there is an n*n matrix of one-forms on AFR(X) such that:
d·1 - «? Λ β . <5.13)
i j 2
Thus, (u ,u,) forms a collection of n + n - dua(AFR(X)) one-forms on AFR(X).
Let us see how these forms transform under a transformation ♦ € G = AFF(X)
of the form (5.1).

482
APPENDIX 2 - METHOD OF THE MOVING FRAME
φ*(Ρ)(χ' χ, χ )) - P(Ax+x_> Ax.,...,Ax )
Ax + x0
(AP + χ.) (χι x,,..- ,x )
ο ι π
♦ *(P) - АР + К . <5.13>
Hence»
d**(P) - Ad P
- Α (ω α β )
- (ω1ΑΑ(«1))
- ♦ *№)
- ♦*(wiAei>
- ♦*(ω1) a 4*(ei) (5-14)
Theorem 5.3. The one-forma (υ ,ω ) are linearly independent at each
point of AFR<X), hence define an abaoluta parallelism. They satisfy the
following equations, the Cartas Moure r equations for the affine group:
^1 1 J
ow ■ ω, α ω
i i k
**J " "k A "j
(5.15)
In Cart an'β tern, they are the atmatiav equations for affine geoaetry. An
affine aomeation for X (in Cart an'β sense) is now defined by giving a fiber
space
πι Ζ ·*· Χ
with base X
2
dim Ζ - η + η
- dim AFF(X) ,
and an absolute parallelism for Z, defined by one-for»·
i i

APPENDIX 2 - METHOD OF THE MOVING FRAME
483
such that
(5.16)
ou - ω, Α ω + Τ, ω Α ω
л i i к i к I
where
he foliation ы -Ο has as leaves the fibers of the nap w.
6. THE FRAMES AND STRUCTURE EQUATIONS OF PROJECTIVE GEOMETRY AND PROJECTIVE
CONNECTIONS IN CARTAN'S SENSE
Now, let Ζ be an (n+1)-dimensional real vector space. Let X be the
projective space associated with Z, i.e., X is the space of one-dimensional
linear subspaces of Z. Let PFR(X) be the space of (n+1)-tuples
«Vi n> (6Л>
satisfying the following condition:
ΙηΑ"·Αϊ ί О (6.2)
ο η
i.e., ел>...,с form a basis for Z.
ο η
Let
wi PFR(X) ·*· X (6.3)
be the map defined by the following formula:
t(i.,...ri ) - (r) = the one dimensional linear subspace
П (6.4)
of Ζ spanned by r_ .
Choose the following range of indices and summation conventions:
1 <_ i,j <_ η .
Let (ρ0'Ρ|) be the maps: PFR(X) -*· Ζ defined by the following formula:
vvi ν ■ ro
Wi V ■ ri
(6.5)
Here is a basic result, whose proof is left to the reader:

484
APPENDIX 2 -- METHOD OF THE MOVING FR*E
Theorem 6.1. There is a basis of one-for»· on PFR(X) labelled
. О i О О i i.
(ω ,ω t« »«i#« r«.)
satisfying the following conditions!
0 i
dP« " w„P„ + ωΛΡ,
0 0 0 0 i
(6.6)
*i " "iP0 + WiPj
These fores (vhich are essentially the Cartan-Maurer forme for the Lie group
GL(n+l,R)) satisfy the following structure equationst
<
<
<
4
-
-
-
-
0 i
U. A Ui
i 0
0 0 0 j
ω Α ω, + ω, Α ω.
i j
"j A ω0
i 0 i к
ω. Α ω. + ». Α ω.
(6.7)
Let us determine the (ooapletely integrable) Pfaffian eyetee whose leaves
determine the fibers of w.
6.2. Let t-*-o(t), 0 < t < 1 be a smooth curve in the fibers of w.
Then, there is a smooth curve t -*» a(t) in R such thatt
— P„(o(t)) - a(t)Pfo(t)) . (6.Θ)
dt 0 0
Proof. That t -*- o(t) stay· in the same fiber of τ means that there
is a smooth curve t -*■ X(t) in R- (0) such thatt
P0(0(t)) - X(t)P0(o(0)) (6.9)
Differentiate (6.9), and define
a - λ — (6.10)
giving (6.Θ).
Now we can feed (6.Θ) into the structure equations (6.6) to give another
form to the conditions that the curve t -*■ o(t) be in a fiber of vt
a(t)P0(o(t)) - «°(£) P0(o(t)) + «*(£) Pi(o(t)) (b.ii)
whence:

APPENDIX 2 - METHOD OF THE MOVING FRAME
485
(6.12)
Conversely, if t+ o(t) is a curve in PRF(X) such that
(ι) ~^ ■ О
О dt
the steps are to prove that t -*> a(t) lies in a fiber of w. Hence,
we have provedt
Theorem 6.2. The Pfaffian system
«J - 0 (6.13)
is completely integrable (this is checked by the structure equations (6.6)1),
and defines a foliation whose leaves are the fibers of w.
We now have all the information needed to generalize (as Cartan did, at
least modulo modern fiber bundle and calculus-on-manifold theory, in [23]).
Definition. A projective connection (in the sense of Cartan) on an n-
dimansional manifold X is defined by the following data:
a) A manifold Ζ
b) A submersion map t: Z> X
c) An absolute parallelism
, 0 0 i i.
on Z, such that
d) ω_ - 0 is completely integrable and defines the fibers of w
e) There are functions
/о0 о0 о1 о1 \
(ROij' Rijk' ROjk' RjM
on Ζ such that the following conditions are satisfied:
. 0 0 i 0 i j
*?0 ■ *i A % + ROij % A %
. о о о о j
-l ™ S A !?< + !Ϊ4 Α !?<

486 APPENDIX 2 - METHOD OF THE MOVING FRAME
*ίο * Sj A*o + Rojk U A*o (6Л4>
. i i 0 i к i к t
^j " ΪΟ A ?j + fti A Sj + Rjk« ΪΟ A "0
Final Remarks. I hope I have don· enough translation of Cartan's Methods into
modern calculus-on-manifolds notation so that the reader can go directly to
Cartan's four basic papers [23-26] on affinc, projective, Euclidean conformal
connections. In Interdisciplinary Mathematiae, Volume 10, Chapters 10-11, I
have provided a modernization end further details about the relation between
Rienannian, projective and conformal connections, which is the point I found
most obscure in Cartan's papers. S. Kobayashi [10] has provided a definitive
exposition of this material from the Ehresmann point of view, which is
somewhat different from Cartan's.
Bibliography
1. P. Klein, Das Srlanger Proegvam, Mathemxtiole Abbendiinger, Vol. I,
Julius Springer, Berlin, 460-497 (1921).
2. F. Klein, Hohere Geometrie, 3rd edition, Chelsea Pub. Co., mw York.
3. С Ehresaann, Les connexions infinitesiaales dans un ·space fibri,
Collegua de topologie, Bruxelles, 1950.
4. S. Sternberg, Leoturee on Differential Geometry, Prentice-Hall, Bnglewood
Cliffs, N.J., 1964.
5. s. Kobayashi and к. Nomiau, Foundation* of Differential Geometry,
Vols. I-II, Wiley, H.Y., 1963, 196Θ.
6. N. Hicks, Notes on Differential Geometry, van Hoatrand, 1965.
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Nordhoff, 1976.
Θ. e. Bishop and r. Crittenden, Geometry of Motif olds. Academic Press, 1964.
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Preen. N.Y., 196B. 2nd edition. Math Sd Pr*ee, 1977.
15. N. steenrod. The Topology of Fiber Bundles, Princeton Univ. Press, 1950.

APPENDIX 2 -- METHOD OF THE MOVING FRAME 487
16. J. Hilnor and J. Stasheff, Characteristic Claeeee, Princeton Univ. Press,
1974.
17. G. Ricci and T. Levi-Civita, Kathodes de calcul differential abaolu et
leurs applications. Hath. Am, 54 (1900), translated in Lie Groups:
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Fr. 7B, 65-127 (1950).
20. E. Cartan and J. Schoutan, On the geometry and the group-manifold of simple
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76, 33-65 (1954).
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1960.
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Ъ2_, 205-241 (1924).
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relativity genlralistie. Annates, Ecole Normale, 40, 325-412 (1923)
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Appl. 2, 167-192 (1923) {Oeuvres, Pt III, vol. I, p. 633).

Appendix 3
FURTHER REMARKS ON CARTAN CONNECTIONS
1. INTRODUCTION
Th· theory of Cartan connections is a beautiful unification and
generalization of classical differential geometry, which deals with the Riemannian
projective and conformal differential geometries in a more-or-less unified way.
It can also be used to treat such non-classical geometries as the "Cauchy
Riemann" (CR) structure on the boundaries of open sets in complex analytic
manifolds. In this appendix» I will review certain key examples, ideas, and
calculations.
Let *-. Ζ -» X be a submersion mapping between manifolds and let
ω1 - 0 , i - 1,
η - dim X ,
be a completely integrable Pfaffian system defined on an open subset U of Z,
where leaves are the fibers of * restricted to U. (Thus, ω € Φ (U),
In d
and the ω (ζ),— ,ω (ζ) are linearly independent elements of X at each
point of U.) The key question ist When does there exist a complementary
set of one-forms ω in U, which form a basis for one-forms in U, and
which are, in some sense, "intrinsically" attached to the ω ?
I will not attempt a full-scale treatment of this question here: Some
developments are in my peper "Cartan Connections and the Equivalence Problem
for Geometric Structures".
2. THE CARTAN LEMMAS
These lemmas are simple algebraic remarks (which just involve linear
algebra) that Cartan uses repeatedly in his work on the Equivalence Problem.
Theorem 2.1. Let г be a manifold, ω, l<i,j <n, be a set of
everywhere independent one-forms on Z. Let Ы.) be a matrix of one-forms such
that
ω* A tJ - 0 (2.1)
Then there are functions f,. such thatt
wj " ^k** <2·2>
489

490
APPENDIX 3 - CARTAN CONNECTIONS
Proof. It suffices to work locally. Let (ω*) , n+1 1*1 din Ζ be a
complementary set of one-for»· such that (u ,ω j form a basis of one-forme.
Then, there are functions (f7 ,f7 ) such that:
jit ja
J: - f^u* + f* ω* . (2.4)
Insert (2.4) into (2.1). since the ω л ω and u лн are linearly
independent, this forces the following condition!
fT ы α ω - 0 ,
3*
which implies ι
f\ - 0 . (2.5)
Insert (2.5) into (2.4), and then into (2.1) agaim
f* ы* л ω* - 0 , (2.6)
which forces (2.3).
Theorem 2.2. With the hypotheses of Theorem 2.1, suppose -in addition that ι
^+ чА ■ ° t2-7>
where (9"<k> i* * eyimetria, non-degenerate matrix of functions on Z. Then,
ω - 0 .
Proof. Set
"ij " «Lk-j (2·β>
Then, (2.7) is equivalent to:
ω + ы L - 0 . (2.9)
It follows from (2.1) that:
ω AUj - 0 (2.10)
By Theorem 2.1,
Uij " fijk"k '

APPENDIX 3 - CARTA* CONNECTIONS
491
with
fikj " fijk · «■">
(2.9) iaplies thatt
Let us combine (2.11) and (2.12) successively:
fijk ' fiXj ' " ^j ' " *kji ' fjki ' fjiX ' " fijk '
whence
"m. ■ ° ·
Q.E.D.
Theorem 2.3. Let (ω ) again be a set of independent one-fогив on the
manifold Z. Suppose (6 ) are a set of two-forms on Ζ such that
в.А (Л...АыП) - 0 . (2.13)
Than, there is a unique set of one-forms ί*»^) on z such that:
6i " "ii A "^ (2.14)
uij - -"ji · «■">
Proof. (2.13) implies (from linear algebra again) that there are
functions f such that:
9i ■ fijk ωί Α W>C ' (2.16)
fijk - - fuj (2.17)
Let us look for ы of the following formt
"ij " ^jk^ (2·1β>
hijfc - - hjiX (2.19)
(2.14), (2.16), and (2.18) require that
fijk " 2 thijk-hiXj> ' ί2·20>

492
APPENDIX 3 - CARTA* CONNECTIONS
what we must now show is that (2.19)-(2.20), considered as linear inhomogeneoos
equations for h , have a unique solution. The argument of Theorem 2.2
implies that the corresponding homogeneous equations have only the zero
solution. Since (it is easily checked) there are as шалу equations as unknowns,
the inhomogeneous equations (2.19)-(2.20) indeed have a unique solution.
Remark. One nay find the explicit solution to (2.19)-(2.20) by successive
permutation of the indices. This is left as an exercise.
Here are two classical applications of these algebraic results.
Theorem 2.4. Let Ζ be an n-dimensional manifold with a Riemannian metric
defined as a positive definite inner product
tvl'V2} "* < Vl'V2>
on tangent vectors. Let (ω.) be a basis of one-forms on Ζ which is
orthonormal relative to the inner product < , > , i.e. t
<Vl'V2> " uitvi>"itv2>
Then there is a unique matrix of one-forms tw,^) such that:
du - ω., Α ω, (2.21)
ω + ω.± - 0 . (2.22)
The Levi-Civita affine connection (in the Koszul sense [ ]) V associated
with the metric is defined as follows:
"i^ViV ' VW* " "ij^V'VV * t2'23>
For more detail about this approach to the connection theory associated
with Riemannian metrics, see Differential Geometry and the CaUmlue of
Variations, i.e.. Interdisciplinary Mathematics, Volume 17.
Here ie another application.
u
Theorem 2.5. Let X be a submanifold of R . Let Ζ be the bundle of
Ν Μ
orthonormal forms of R which are tangent to Z. Let Pi Ζ ■» R be the
H
map which assigns to each frame the point of R to which it is attached.
н
Let (e ) t 1 <_ irj <_ η - dim Ζ be the R -valued functions in Ζ which
represent the value of the i-th vector coordinates of the frame. Then, there
are a baeie of one-forms Ζ labelled
(ω.,ω.)

APPENDIX 3 -- CARTA* CONNECTIONS
493
wij + wji ■ °
such that
dP - ωιβι
Ал ш Ш Л
i ij j
(2.24)
(Relations (2.24) are to be considered as relations in the exterior algebra
of R -valued differential fores on Z, in the sense described in Appendix 1.
Of course, Cartan himself, in the book, essentially proves Theorem 2.S.
Now we shall attempt to generalize some of these ideas.
3. TORSION TENSORS
Let
w: Ζ -*- X
be a submersion map between manifolde. There are then two vector bundles on
Ζ which play a basic role in the Cartan-Ehresmann connection theory, the
bundle of tangent and traneVereal tangent vectors to the fibers of ,
described ast
χ
Ε , Ε
π π
Ε - ((г,ν): ζ£Ζ, ν€Ζ , π (ν) - 0) (3.1)
W Ζ "
χ
Ε - quotient vector bundle of T(Z) by the subbundle
Ε defined by (3.1)
(3.2)
Definition. A torsion tensor associated with the submersion map-fiber space
(Ζ,π,Χ) is a bilinear bundle map
XX X
Τ: Ε Φ Ε ■♦ Ε (3.3)
WW W
such that
T(v1#v2) - - T(v2,Vl) (3.4)
for vj/v2 e Et
X X
(In other words, Τ is a linear map of the exterior product bundle Ε л Е
into Ε .)

494
APPENDIX 3 - CARTA* CONNECTIONS
Let us s«e what this mtans in teres of bases of one-forms on Z, i.e.,
'moving frame·" in the sense of Cartan. Suppose
η - dim X
1 <i,j,k < η .
(ω ) is a set of independent on*-fones such that the
Pfaffian system
is completely integrable, and the fibers of w over
the leaves of the foliation.
Then ι
ω - 0 on the fibers of Ε , hence w passes to the
* χ
quotient to define linear forms on the fibers of Ε ,
-i
which we denote as ω . Our assumptions imply that the
-i ·*■
ω are a basis for the dual space to each fiber of Ε .
χ i
Let e denote the dual basis Ε . Then, и defines
i w
canonically a "moving frame" e as a cross-section of
j.
the basis bundle to 2 .
t
Given a toreion Ыпвог Τ defined by (3.3), we can now define its
oonponmnta with respect to the moving from as the function (T J) on
Ζ such that
i(W - *ij\ · (3·5>
In order to see where the name "torsion tensor" comas from* let us recall
tho definition of the toreion termor of an afffine oonneation in Xommul 'β
ββηββση X. If the affine connection is defined by an R-bilinear map
(VV * \ V2
of y(x> * V(X> ■* V(X> euch that
V V ■ £7 V
fV 2 "νχ 2
for f e 04x) , νχ,ν2 e y(x)
\|ДУ ' Vf)V2* *\У2
The toreion tensor (of the affine connection) is then

APPENDIX 3 — CARTAN CONNECTIONS
495
T(vrv2) - γ2 - -^ - [vl(v2] (3.6)
Suppose that
(V
is a basis for vector fields on X. Then
T(Vi'V " ^L}\ (3"7>
where (TV.) «re functions on X, which are the oorrponenta of the torsion
teneore T.
In Cartan's theory those TV Are the pull-back of a cross-section
where Ζ is the bundle of tangent vector bases of X»
and where the (TV ) are defined by relations like (3.5).
However, I do not mean to pursue this "elegant" algebraic approach to the
torsion tensor, but to follow Cartan's more intuitive path.
4. THE TORSION TENSOR AND CARTAN'S STRUCTURE RELATIONS
Continue with (Ζ,π,Χ,ω ) as in Section 3. The (ω ) are one-differential
force on the manifold Z, hence can be exterior-differentiated. (This is
always Cartan'· approach!) Suppose that there are one-forms ω on Ζ and
к ^
functions (T,.) on Ζ such that
du ■ ω. л ω + Τ ιύ л ω . (4.1)
j J*1
Of course ι the (ω.) do not have to be independent nor do the/ have to
define a set of horizontal forms for w in Ehremann's sense. However, we can
investigate their indeterminacy with the aid of the Cartan lemmas discussed
in Section 2.
Suppose that, the Τ лхо the cumpuiiantu of an "intrinsic" tensor field
xx χ
Τ: Ε Λ Ε -*■ Ε ·
as defined in Section 3. Suppose that
(йЬ

496
APPENDIX 3 - CARTAN CONNECTIONS
is Another choice of one-fore» which satisfies relations (4.1). Then,
(ω, -ω.) л ω
(4.2)
Hence,
-1
i i к
uj + Yjku
(4.3)
with:
"jk
i
Ykj
(4.4)
Relation (4.3) then tell us how to change the ω , while keeping the
"torsion" part of the structure equations, namely, (4.1), unchanged. In the
many examples Cartan worked through, he started off with a choice of "T" (most
commonly.
Τ - 0), then chose the (Y..) so that ω, had some sort of
oanemioal form. Then, he exterior differentiated the normalised ω , and
5. THE COMPLETE STRUCTURE EQUATIONS
Continue with the notation of Section 4. The - structure equations are the
following set of relations:
du
duT
du
j*
* J- J *
л ω + Τ ω л ω
J*
к iii k i к 1
. л ω + ω,. * ли, + т.. . ы л ω
к gk ΐγ jki
(5.D
(5.2)
Ι ϋι 1 iili9 * «i *1 *-2
. л ω + ω,, ι л ω. + ω,. ; ζ л ω. + Τ... . u * л ы '
kt jkl ij jkl ijij jkli*2
(5.3)
and so on.
Again, there are certain changes in the ω., ω.. , w,* 2,
3 Э* J*
which leaves
these structure equations unchanged. (The physicists call them "gauge
transformations" 1)
The grand strategy is now to classify the "generalized geometries"
(affine, Euclidean, conforraal, projective, —) by ям ana of special ohoioes of
the relatione satisfied by these fores. I will not be able to give a complete
analysis here, but will restrict attention to certain special situations.

APPENDIX 3 - CARTAN CONNECTIONS
497
6. GENERALIZED AFFINE GEOMETRY
Suppose the structure equations take the following form:
dw ■ ω, л ω + Τ,. « л ω (6.1)
J J*
Ai i к iii к J. к I ltL _t
dw, ■ ω,, л ω + ω,.1 л ω, + R,.. ω л ω (6.2)
j jk jk lj jki
where the following conditions are satisfied:
The on·-forme (ω,*ω?) axe linearly independent at ., ..
11 (ο·3ι
each point of Ζ
ω,. 1 is a linear combination with constant coefficients . ..
jK ■ (6.4)
of th· («,*w?).
Since the Pfaffian system {ω J is completely integrable, and the leaves
are precisely the fibers of v, the ω,, restricted to each leaf of w,
3 2
satisfy the Caxtan-Maurer equations of a Lie group of dimension η . In the
special case that the second tern on the right hand side of Equation (6.2) ist
ω. л ω, (6.5)
and the first term is zero, this Lie group is GL(n,R), the group of n*n
invertible real matrices (or linear automorphisms of R ), which is why we call
this structure "generalised «ffino 9*om*try". The ocrafticionts on th· third
term on the right hand side of (6.2) are denoted by NRN to recall the standard
formulas, where "R" stande for the Riemannian curvature tensor. Thus* the
structure for the (torsion-free) affine connection takes the following form:
dw * ω, л ω (6.6)
dw, ■ ω. л ω, + R, , ω л ω (6.7)
Suppose such a structure is given on Z. Following Cart an, we will define
the development curves in ζ into the affine frame bundle of Rn. Let
t ■+ o(t)
be a curve in Z. Let
do ,.
' * df£Zo(t)
be the tangent vector curve to a. Let

498 APPENDIX 3 - CARTAN CONNECTIONS
AF(R ) - set of affine frames of R
((Pi e.,...,e )i ?>·.(...,« €r (6.6)
1 η Ι η
β, β axe a basis for Rn)
1 η
Associate with σ the following ordinary differential equationai
dP i/ άα\
dt " U (dt β1
(6.9)
dt" uj(dt)ej
We can then see various geometric meaning to conditions that are imposed
on the functions
i/do\ i/do
*-(£)< ·(£)
Theorem .6.1.
i/do
-IS)
(6.10)
i.e., t -*· o(t) is horizontal with respect to the Ehresmann connection for ν
defined by the (ui,)# if and only if the curves
t ■+ (P(t), ei(t))
in AP(R ) defined by (6.9) satisfy the following condition:
The t -*■ β (t) are constant. Geometrically, this means
that the affine frame (P(t)j e. (t) e (t)) is, for
1 η
each value of t, parallel to the intitial frame
(P(0)j e.(0) e (0)) -
ι η
This gives, in this caae, the geometric interpretation of the Ehresmann
"horizontal" conditions (6.10).
Theorem _6.2. Let
t ■+ (P(t) ι e. (t) e (t))
1 η
be the affine frame defined by the equations (6.9). Then, e1(t) is tangent
to t -*· P(t) if and only if the following condition is satisfied:

APPENDIX 3 -- CARTA* CONNECTIONS
499
Αέ) - · ""(£)
Geometrically, this can be restated as follows:
The integral carves of the Pfaffian system
u2 ..... ип . о (6.12)
develop, into AF(R ), to curves such that the first vector t ■* e (t) on the
frame is tangent to the (unparameterised) curve t ■* P(t).
We can now set up to define straight linma of the affina structure, λ
parameterized curve t ■+ P(t) in R is a straight line if it is of the
following form:
Pit) - a(t)vQ + νχ (6.13)
wh«
Then,
v0,v1 e r ,
dP _ d£
dt " dt V0
2 2
О d q
dt2 " dt2'0
Suppose now that
t ■+ (P(t), e.(t) e (t))
1 η
is also a curve in AF(R ) determined by Equations (6.9), subject to the
tangancy condition (6.10). Thus*
/do\ ,^ dP
iao\ ,„. dP
-l(dtK(t) Ш Λ
£vQ . (6.14)
Differentiate both sides of (6.14), and use relations (6.9) ι
$*o -Ie (lift)) ·!*-!(£)&
- It "ι(λ) ·ι + "ι(|Ι)("ί(*)·ι) ·

500
APPENDIX 3 - CAHTAN CONNECTIONS
Comparing both sides of the equation, and assuming that
do
-l(dt) ' ° ' (6'15>
we see that:
"1(f) ■ о ..... -Я*)
Tbus, we see that we have expressed the conditions that the development
of σ in AF(R ), subject to the tangancy conditions (6.9), be a straight
Una in terms of the affine geometry of R . (ftien, the straight lines form
a family of curves in R which is preserved under the group of affine
automorphisms.) Me can then carry over this family to X, the base apace of ζ
by the projection map wt ζ -*■ X.
Definition* A curve in X is л straight tin* (on the affine connection
geometry defined by the forms (u ,»,) on Z) if it is the projection under
w of a curve in Ζ which is an integral curve of the Pfaffian system
л 2 η 2 η . ,_.
0 «ω ш -·· ш οι "u. ■·-·■(!! . (6.17)
Let us now determine the condition that two (torsion free) affine
connections have the same straight lines. Suppose the first connection is
determined by forme
Ш ,(i), ) ,
the second by forms
(ω ,ω )
with:
"i " Ui + Yik "* [6.18)
Y*k - YKi · (6Л9)
Tftie condition that the two connections determine the pure straight line
is then that the forms
■2 η ._ __.
ω. * ..-»ω. (6.20)
be a linear combinstion of the forms:

APPENDIX 3 - CARTAN CONNECTIONS
501
2 η 2 η ., ...
ω ,...,ω , ω. ,...,ω. (6.21)
The condition can then be read off from (6.18), namely:
j - j 1
ωί = Ύιιω
modulo the forms [6.21), for j > 2. Hence, the condition that the two affine
connections determine the same straight line is that:
y\x - 0 (6.22)
for j >_ 2 .
Definition. Two connections which satisfy (6.18)-(6.19) and (6.22) are said
to be projectively related.
One can now describe how the curvature tens ore of two projectively related
affine connections are related. This calculation--and the calculation leading
to the projective curvature tensor is done in detail in Vranceann's treatise
This is the approach that Cartan takes toward defining the straight lines
and the projective changes of affine connections.
7. THE MOVING FRAME FOR PROJECTIVE GEOMETRY
Now, start with R and construct the spaceι
Ε ■ {(Pi e. e ) : P, e_,...,e € Rn j |рлв, л -·- ле 1-1}
in in ι η
(7.1)
(| I is, here, the structural L_-norm on tensors of R . This condition
|рлел---ле | - 1 means, geometrically, that the volume of the subset
(τΛΡ + т.е. + ··■ + τ e : 0 < τ.» τ. τ < 1}
О 11 ηη — 01 η —
of R is one.)
Let Ρ (R) be the η-dimensional real projective space, defined as the
one-dimensional linear subspace of R . For Ρ С R , Ρ ψ 0, let [Ρ]
be the one-dimensional linear subspace it generates. Let
w: И ■* Ρ (R)
η
be the map defined as follows
w(Pi e.,...,e ) - [P] . (7.2)
1 η

502
APPENDIX 3 - CARTAN COWIECTIOKS
Now, consider Ρ and β , 1 <_ i,j <_ n, as R - valued sero-th degree
differential forms. The structure equations axe then:
i 0
dP ■ ω β, + ω Ρ
J ^ °-
ωτβ. + ω.
(7.3)
de в и е + ш Р
1 i j i
where (u (u.(u §ш~) are a basis for aoalar one-forms on Z.
Now,
d(PAe,Λ···Λβ ) ■ (w + w +--·+ω )Рле,Λ-.-лв (7.4)
1 η 0 1 η 1 η
Hencer the volume preserving condition
|рле л ·-- ле I - 1 (7.5)
for the moving frases on Ζ requires that:
ω + ω ■ 0 (7.6)
λ curve
t ■+ (P(t), ei(t)) - a(t)
in AF(Rn+ ) is on the fiber of w if and only if t ■* P[t) lies on a one-
dimensional linear subspace of R .In view of (7.3)» we have
ut " - Ш\ + " (dt)p (7·71
i da
ω dt " ° (7'8>
Thus, the fibers of w are the leaves of the Pfaffian equation:
ω1 - 0 . (7.9)
Let us check that the equations (7.9) are aoeptetely integrabl*. For this, we
must find the structure equations satisfied by the forms (w ,«?»w #«,)* we
can do this by exterior differentiation on both sides of (7.3):
i 0 i i 0 OiO
0 ■ due+dttP-w л (w*e + u Ρ) - ω л (w β. +« Ρ)
0 ■ dwje + dui.P - «J л (ω β + ω Ρ) - ы. л (*re. + w Ρ)

APPENDIX 3 - CARTAN CONNECTIONS
503
whencej
Αι1 - J л nS в° л ω1 (7.10)
.0 i О #111»
άύ - ы л ω, (7.11)
j к j 0 j
*"i " »ϊ Λ «ί + »i л « (7.12)
л о о о ,„ ,,.
do^ ■ ы. л ω . (7.13)
(7.10)-(7.13) are the Structure Equations of Projective Geometry, ibey are
also the Cartan-Maurer equations for the Lie group SL(n+l, R) ι However,
their great virtue (at least from the point of view of doing moving frame
calculations) is that they are automatically adopted to the coset space
SL(n+l,R)/G - Ρ (R) ,
η
where G is the subgroup of GL(n+l,R), which is the i sot ropy subgroup at
one point of Ρ (R). In fact, G is the subgroup defined by the Pfaffian
η
equation (7.9)* which, in fact, is completely integrable, as we can see
from (7.10).
Having written the Cartan-Maurer equations SL(n+l,R) in a form
adapted to the coset space SL(n+l,R)/G ■ Ρ (R), we can now define a projec-
n
tive connection by Cartan's method.
Definition. Let Ζ and X be manifolde with:
dim X ■ η
dim Ζ - (n+1)2 - 1
- n2 + 2n
■ n(n+2j
2
■ η + η + η
Let ν: Ζ -» X be a submersion mapping, λ projective connection for (Z, ,X)
is defined by giving an absolute parallelism on Z, defined by one-forms
labelled
(ej.e^ej.e0 - - e*) , (7.14)
satisfying the following structure equationst

504 APPENDIX 3 - CARTAN CONNEaiOHS
de1 - ej л e* + e° л θ1 + τ* θ^ λ θ* (7.15)
de° - e1 л ej + т° ej л ек (7.16)
dej - e* л ejj + ej л ej + *? ek л θ* (7.17)
dej - ej α ω° + TJjk ek л θ* (7.ιβ)
The functions (T,R) by which the Cartan-Maurer equations for SL(n+l,R)
are modified are the toreion and curvature for the projective connection.
Given a curve t ■* o(t) in Z, we can define its development in the
space E. They are the curves defined by the following linear ordinary
differential equations:
dP -i/do\ A0/da\ _
dT ■ θ (it) ei + θ (it) p
(7.19)
dt
г - βί(£) ·,+ i(S)p
Me can then project the curve σ down to X via the subnersion map
the map:
w: Ζ -*■ X, and similarly project the solutions of (7.19) down to Ρ (R) via
(P. e.,....e ) -*■ Ρ .
1 η
In this way we obtain a development (better, a "correspondence") between
curves in X and curves in Ρ (R). The torsion and curvature (T,R) enters
η
when one develops homotopies. i.e., surfaoes.
I will now indicate how these idea· fit into the Ehresmann jet-bundle
formuas.
8. CARTVJ'S METHOD OF THE MOVING FRAME AND EHRESMANN'S JET CALCULUS
In principle, Cartan'e method of the moving frame, and associated Cartan
connection ideas, can be squeezed into the Ehresmann jet calculus. However,
in practice, the fit is not that precise, and many details have not been
worked
As I have already indicated, what is beautiful about Cartan'e method is
that the formulas are not imposed abstractly, but are worked out
recursively. All spaces are embedded into products of vector spaces, so that
curves can be described in terms of linear differential equations, much as
curves in R are described by the Frenet-Serret formula. (The coefficients

APPENDIX 3 -- CARTAN CONNECTIONS
505
of thee· equations are the "geometric invariants" of the curve. Ση the case
of R , they are the classical curvature and torsion of the curve.)
Let χ be an n-dimensLonal manifold on which a Lie group L acts
transitively. Let χ be a point of X, with G the ieotropy subgroup of
L at χ . Thus, X may be identified with the coset space
L/G .
Let
jr(X.Rn>
be the space of r-jecta (r _> 0) of mapat X -*■ R . The action of L on X
prolongs to an action on J (R ,X) ,
Lx jr(X,Rn) ■+ Jr(X,Rn) (8.1)
Let 4 be a mapi X -*■ R , such that
4(x0> - 0 (8.2)
♦.' X0^RS (8'3>
is an isomorphism. Let
j - η-jet of ♦ at the point χ (8.4)
Ζ - orbit of ^ at j (8.5)
Suppose r is chosen so that the following condition is satisfied:
L acts in a neighborhood of j in J (X,R )
effeatively, i.e., it lj - j for I E L, j In thie
neighborhood» then 1 - identity element of L.
Let
wj Ζ ■* X
Ь* the canonical projection maps J (XrR ) -*■ X (which Ehroamann called the
target map) restricted to Z.
To get Cartan'β machinery working» we need something more: A vector
space Ε of dimension n, a faithful linear representation
p: G ■* GL(E)
of G by linear automorphisms of B, and a map

506
OS jr(XfRn) -»■ ε"
APPENDIX 3 -- CARTA* CONNECTIONS
(E - Ε * ··· « Ε - Cartesian product of
ing conditions are satisfied:
|o(Jn>| * 0 .
μ copies of E) such that the follow-
(8.6)
α intertwines the action of L on R (X,R )
and the linear action via ρ of L on Ε .
(8.7)
α restricted to the orbit of L, i.e., Z, will then define m
E-valued, cero-th degree differential forms on Z. The scalar one-form
coefficients of the external derivative of this saxo-form will then define
the Maurer-Cartan form of L, but in a way that is adapted to the "geometry"
of the coset space L/G - X.
9. THE METHOD OP THE MOVING FRAME AMD THE INFINITE DIMENSIONAL LIE GROUPS
OF CARTAN
As a final remark, note that one can also consider the case
and in this way cover the situation where L is one of the "infinite" groups
considered by Cartan in the second part of his Oeuvree. In these "Infinite
Lie Group" papers he, in effect, wrote down the "moving frame-Maurer-Cartan"
equations for these objects* which are, of course, рвлиОО-дтоирв in the sense
of Ehresmann, and have been extensively studied (but not from this point of
viewI) by D.C. Spencer and his collaborators.

INTERDISCIPLINARY MATHEMATICS
BY ROBERT HERMANN
1. General Algebraic Ideas
2. Linear and Tensor Algebra
3. Algebraic Topics in Systems Theory
4. Energy Momentum Tensors
5. Topics in General Relativity
6. Topics in the Mathematics of Quantum Mechanics
7. Spinors, Clifford and Cayley Algebras
8. Linear Systems Theory and Introductory Algebraic Geometry
9. Geometric Structure of Systems-Control Theory and Physics, Part A
10. Gauge Fields and Cartan-Ehresmann Connections, Part A
11. Geometric Structure of Systems-Control Theory, Part В
12. Geometric Theory of Non-Linear Differential Equations, Backlund
Transformations, and Solitons, Part A
13. Algebro-Geometric and Lie Theoretic Techniques in Systems Theory,
Part A by R. Hermann and С Martin
14. Geometric Theory of Non-Linear Differential Equations, Backlund
Transformations, and Solitons, Part В
15. Toda Lattices, Cosymplectic Manifolds, Backlund Transformations and
Kinks, Part A
16. Quantum and Fermion Differential Geometry, Part A
17. Differential Geometry and the Calculus of Variations, 2nd Ed.
18. Toda Lattices, Cosymplectic Manifolds, Backlund Transformstions and
Kinks, Part В